Nice 'study', but where is Tower?

I personally disagree with most of those "guides". 1st of all because I don't believe in those theoretical tiers.
If you add up the cost of each upgraded units from both Castle (which you think is in the top 3) and Dungeon ("the bottom tier") you end up spending more golds with the Castle.
I do agree that Castle is much easier to build up than Dungeon or even Stronghold (which require sulfur and crystal that can be tricky to find sometimes) but that's not enough reasons to make Castle any better than Dungeon or whatever.
Just because dwarves and dendroids are slower units doesn't make Rampart a "middle tier". Faster units are not better just because they are fast. This is just a tactical choice. Defensive creatures are as valuable as fast units. I personally love the dwarves and their 40% magic resistance.
Plus, Rampart had the fastest 1st level creature before Conflux came out. It also has Pegasi (hp issue I agree with that) and gold dragons that are pretty fast.
Don't take offense but to me, those tiers and theories are non sense.

This.
Also "nferno and Fortress are at the bottom because their units are unnaturally weak - Fortress lack a decent shooter......." is a utter bullshit. On small(er) maps with high difficulty those "top tiered", expensive castles will get owned by "inferior" Fortress, which will come visit their town with wyverns and such (yes, they can be built VERY fast).
This is the same as in HoMM2, where Knight and Barbarian castles are considered inferior. They just shine on smaller maps with fast attacks. Same as Fortress.

Agreed with the HoMM2 quote, but if you think Fortress can get over their lack of a poor shooter then you're vastly mistaken.
Fast building of Wyverns isn't something that's unique to Fortress. Conflux can get all their troops in the first week, gold permitting. Stronghold can get Behemoths in the first week. You can get Nagas - the no retaliation makes them better than Wyverns - in the first week. Necropolis can pick up Death Knights in the first week.
You might be able to clear out some low level troops with Wyverns and Serpent Flies, but you won't be winning any long term battles. Any decent spellcaster can put a Blind or a Slow on the Wyverns. If their hero has Tactics as well, you've gotta hope that you completely outnumber the opposition in order to survive crossing the battlefield.
Saying that Fortress has an advantage over other castles on smaller maps is complete rubbish. Other castles can break out just as easily; there's a lot more to winning than just being able to build Wyverns on Day 4.
I agree that faster units aren't better just because they're fast. However, in the case of the Dendroids, they're not slow - they're geriatrics! Moving at 3 squares a turn (4 if upgraded) is pitiful. Unless you have teleport, the enemy basically has to come to them.
Now that's fine if you're going to use them to block a hole in a castle wall, or use them as defenders for your Grand Elves but the problem is you lose a lot of speed exploring the map just by having them in your army. On top of that, if your archers get nuked or negated (Blind, Forgetfulness) then you're going to need the Dendroids to do some damage.
Compare the Dendroids to say, Ogre Magi. Or Vampires. Or Mighty Gorgons. Or Minotaurs. They're simply just not as useful in battle. Speed isn't just a matter of speed - it's a tactical advantage in itself. Dendroids and Dwarves take that advantage away from the player using them, and it's part of the reason why they're not the best.

dippizuka, try playing on higher difficulties. On the easiest it is true that Tower can have nagas in first week. But compare resources needed to build naga building (or Necropolis' death knights) to those you need for wyverns. And that is no small difference when you play for example on expert or impossible difficulty.

With that, you have convinced me that fast units have a slight advantage over slower units.
Absolutely not. Try it in game and you will see no difference whether you have Dendroids or not in your army.
If you are lucky enough to find 10 crystals.
But anyway, my point was mainly to demonstrate that there is no such thing as "tier list".

I thought that hero movement was affected by the speed of the slowest unit in its army? It's been a while since I played, but I thought I remembered this...

I never heard of that before and I tried it when Dippizuka mentioned it, and there is no difference.

I always hate to put up tiers to factions or characters. There will always be a tactic that is broken, but that is useless if you use the right counter. There'll always be maps where one faction has advantage for less reliance on resources or the abundance of the same, even the size of the map matters, because the larger the map, the larger the armies once they clash, and some creatures are just better in larger numbers than small numbers. And then there's always a factor of luck. Getting certain artifact early can lead a hero to save a sizable amount of troops.
What can be done, however, is to rate them by how "player friendly" a faction is. Which faction relies on some tricky tactics and which are straightforward.
By this definition, I'd say Castle is by far the easier. All its troops are pretty straightforward in use. Sure you can effectively increase the damage of cavalry with haste, but that's minor to the overall Castle strategies. Finally, the Archangels are one of the few units you can place anywhere in combat and are tremendously difficult to take down. Straightforward use. Rarely t is worth to waste its turn to try and resurrect troops elsewhere.
Conflux is also pretty easy. From the start you have a level 1 creature with good speed and no retaliation. If your first hero only uses them, you can clear a good way around your castle with a little patience. Then the phoenix have the same mobility advantage of the Archangels and will always make sure you're the first to cast a spell.
The Tower can be a bit tough on costs and the clear usage of its creatures, but having a shooter right from the start, a second shooter in the middle and a third shooter in the end creature tier, clearing wandering monsters becomes way too easy and conserving troops becomes second nature, add to that a strong no-retaliation and you can get to your enemy with fully accumulated unities without breaking a sweat.
Necropolis can accumulate skeletons. But it isn't easy because you have a tough time conserving the bastards. But once you get Vampire Lords you have a pretty decent tank that can take most punishment without losing strength. Once you get Dread Knights you can clear the path easily. The most important is to remember to waste the enemies retaliations on your zombies early on, and once you get the hang of it, you'll be on your way to amass ridiculous amounts of troops. Resurrection at 3rd level helps too. The lack of morale can hurt sometimes, but if you get the morale-nullifying artifact or mass sorrow, it will work in your favor.
Rampart has one of the best shooters in the game very early, and the best shooter protector there is. No other faction has a creature who can do well avoiding fast enemies to reach your shooters, and even though they're criminally slow, dendroids do a great job wasting the enemies important creature's turns. Dependable creatures in the start also helps.
Then comes Stronghold. Even though stronghold is almost as straightforward as the castle in combat, it has one major weakness: Most their creatures are pretty frail. The Stronghold is tricky because you can only use them well if you know how to control the enemies retaliation. Wolves do enormous amounts of damage with their high attack and their dual strike, but the retaliation to the first attack usually decimates them. In the start you have to learn to sacrifice goblins at essentially everything. Goblins already come in great numbers and they don't do a whole lot even when their numbers start to swell, but the loss of wolves can costly. Later on the Ogre Mages are absurdly sturdy melee units that must be used with care in order to make the best out of them.
Dungeon has one great strength: Black Dragons, and one great weakness: It takes too damn long to get them. The known "Armageddon + BDragons" combo notwithstanding, playing with Dungeon is an uphill battle against your treasury. Your mid-tier creatures are expensive and your top-tier creature is expensive. And Troglodytes do a sucky job trying to hold off the enemy lines early on, and your true tank only comes at level 5. Then again, minotaurs are so incredibly strong things go smooth once you add them to your ranks. Really, the worst part about the Dungeon is that they have 2 shooters who are neighbors in the tiers, and that makes it lack a strong melee unity to keep you going in the early game.
Fortress has a problem in that in order for it to work well in combat you need to hope that two abilities with a chance of happening do happen: The Death Gaze and the Petrification. If the Basilisks manages to petrify some annoying or powerful enemy unit, the Fortress is at a strong advantage. And if the Gorgons manage to do their Death Gaze upon the 7th tier or 6th tier of the enemy, you're also safe. But if neither happen, you're kind of screwed. The Hydras are pretty strong, but their lack of movement and large size prevents it to have the mobility to deploy well its attack. And in the start the they're pretty lacking, as neither the Gnolls nor the Lizardman are particularly strong, and the Sepernt Flies are too expensive and fragile to protect either.
Finally, there's the Inferno. The greatest problem with the inferno is that its first two tiers are essentially sacrificeable from the beginning to the end. Imps are only good at getting themselves killed, and Gogs can't stand two rounds with a decent melee at their side. And Magogs are difficult to use. Way difficult. Also, like the Stronghold, your duty is to preserve the dogs, but here they're hellhounds. Fortunately, they already ignore retaliation. You must make sure the efreet is the one getting the strong retaliations, because they're effectively making the enemies retaliate against themselves in the process, but they're too expensive to have such an expendable role. The strength of the Inferno lies in the good use of the Cerberus and Pit Lords, as you must conserve the later in order to have effectively 8 stacks in the battle. Demons are mediocre at their role, but at least they're not as proficient as the imps at effing themselves, so they're good to defend your pity lords and let them dish out the real damage. Finally, the Devils are sort of strange. They're weak for a 7h tier, but unlike the bone dragon they don't have a good debuff and unlike the phoenixes they don't have extra growth to make up for it. Their mobility is good, but Archangels are more mobile and cheaper (in a way). Finally, they're very expensive. What gives? Each 7th tier has usually a specific strategy to use them well, but the Devil boils down to a mobile tanker, and one that won't stand its ground well to other tier 7. So, effectively, every creature in the inferno have a tough strategy involving their use, and by my definition, that makes them the worst tier.
Still, this is just my opinion, and facing the Inferno and having its pit lords in full strength successfully summon their demons is not amusing in the least, as well as noticing you've done something stupid and see him use his mobile cerberus to effectively act as 3 creatures that won't be punished by their attack. Even worse when it is twice (smart Inferno players wait the cerberus turn so it can effectively attack twice slower clusters of stacks.

"I never heard of that before and I tried it when Dippizuka mentioned it, and there is no difference."
Read, or download, the Heroes 3 manual.
It specifically says that the faster the slowest unit in your army, the more movement speed your hero gains on the map.

Ok, after further explorations, I have figured something out that I never noticed before. Your slowest unit slows you down on the adventure map only on your native terrain. That is why I couldn't see any difference earlier. Otherwise, there is no penalty at all (you just get the usual effect of difficult terrain that you can compensate with the pathfinding skill). It a little bit confusing but I understand it better now.

Heroes don't suffer any movement penalties on their native terrain, but having slow units in your army will slow you down regardless of what terrain you're on.
If you have pathfinding, it'll negate the movement penalty from terrain but not the movement penalty of your armies. This is only countered by Logistics, or playing the Wake of the Gods expansion (where all the skill-sets are upgraded and Pathfinding increases the minimum movement speed of your army as well as negating terrain bonuses).

This is where I'm doubting. I tried again today 3 times (SoD) with new heroes freshly recruited (no path finding, no logistics), on native terrains there is a difference (big time) but on others terrains there was absolutely no difference. I have tried on lava, rough and grass (couldn't check on snow).

You did not check well, there are difference on all terrains, and it can be massive. It is actually one of the main advantage of the Fortress, it can field Week 1 or Week 2 depending of the level of difficultyy a very quick archer-killer (dragon-fly / wyvern) army to clean up the mines and random monsters.
The number of move points of the hero depends on the slowest creature of his army at the beginning of its turn. Run a random game max size wih weak monsters and a Fortress start. Create and ameliorate a dragon hive ASAP. Start with only a dragon fly in your army, see how far you can go. Then drop the dragon fly and take lizardmen. Conclude.
As for which army is the best one, well, I think that the number of different opinions here show that they rather well balanced the game. I played a lot of Hotseat multiplayer HOMM 3 (about 50, mostly with the same players though), and my prefered castle was the Fortress, including on large maps :
- The dragon-fly is a powerful (expensive, though) creature who will most likely start the combat and will allow you to attack and retrait with the "wait" option in most circumstances. Not very good against a player in combat, but will clean most of your surrounding at the beginning for a great start.
- Gnoll and Lizardmen are poor, granted.
- Basilic is a great unit, since its random petrification will defeat the enemy strategy, including another player. Every "random" effect is great in multiplayer since it allows to break a battle strategy or force the player to cast spell on the unit having this effect to avoid this.
- Gorgons are nice for killing rank 7 units. It will makes the opponent wary of flying his archangels / dragons / whatever just next to your archers, since if he does that he WILL love scores of them due to the Gorgon power. (Lizardmen are the most tranquil archers in multiplayer, since they are weak and not worth losing an elite unit for). Gorgons are also safe from level 7 attack at the beginning of the battle, because the other player might want to use the ripost on goblins or peasants. Of course, the player might blind them. Fine. You can dispel if he does, or do something else.
Note that I believe the psychological effect of the Gorgon breath is way beyond its real efficiency.
- Wyverns... well. We all know how powerful they are.
- Hydras... ok not so good, but with Haste then will be ok.
The Fortress can have the Wyvern VERY quickly, except on the hardest setting. In this situation, the Dragon Fly should be good enough (using the wait attack retreat strategy) to clean quickly enough the surrounding to get that good-damn wood.
On small maps, you will rush your opponent very quickly, on large map or XL map, you ll grab an extra town/castle before they do. Really great.
Now three other general notes on the game :
Having the quickest unit is sure great, but it can be VERY useful in multiplayer to wait before casting a spell. Basically, in MP, both player will understand in which direction the battle will evolve (in most case). Only the pool of spell will break the probable plan of the other player. If you cast a spell first (a powerful spell like Blind), you move the battle in one direction (no archer, for instance), but the other player have the opportunity to either cancel that move (dispel) or to steer the battle in yet another direction, and will have had the "final word". This does not apply to spells like Berzeker, or when a spell allows to attack first and to do a great deal of damage before the opponent does (Mass Haste for instance), but should be kept in mind.
Having the SLOWEST unit in a battle (dendroids for instance) allows a player to make sure he casts his spells in second (except during the first round, since it is likely that a player will use "wait"). In the case of the Rampart, since the Pegasus are so quick, that player will be able to chain cast two spells. VERY efficient, and not to be forgotten.
Having a small cost in "rare ressources" and a normal/high cost in non-rare ressources is IMO much better than having a very cost in non-rare ressources, since in the hardest level you will find at least some rare ressources around, but not enough wood / ore not to have to wait for the mines to ramp up. Of course, having an high requirement in ONE rare ressource is just horrible. Another reason why the Fortress shines IMO.

Ok my bad, I didn't pass my turn so I couldn't notice the difference right away. At least I have learned something new. But anyway, it's not like I care much about that. To me it's a minor disadvantage that doesn't affect the way I play.
That's exactly what I'm saying since the beginning of this thread. Unlike some others turn based strategy games, HoMM3 is a pretty well balanced game. And I keep thinking that there is no such thing as "town tier". You may prefer a town over another for many reasons that are proper to your play style, but the tier ranking is a non sense to me.