Exactly that, I'll edit my post to say that (and add a picture here)

The AI seems very trigger happy with attack spells when running auto-combat with heroes.
But, with the general approach of maxxing out defence, it seems to me that spell points are better spent on healing when available - healing shower, remedy, etc.
The easiest way to force the AI to not use attack spells is to simply not research them.

Question is: do you guys ever research attack spells when you have water or life? If so why? Multiplayer only.

Actually, the more accurate answer to your question is that I don't concern myself with researching most attack spells, and I rarely end up using them in combat because all my spell casting skill is spent on casting buffs and units. My heroes don't have casting skill available for casting combat spells because it has been used (or placed in reserve by holding a spell so it won't be used).

I like chain lightning. First spell I picked and it's been very useful early stages. As was Vaporize and Solra Flare. Now not so useful as his attack got stronger, but still very useful against fliers. I got 2 spheres in Air and one in Life.

When do you learn new spheres?

Any tips on what should I pick next?

I took Life for healing only to note that most healing spells are actually Water. Air was taken for damage. To be honest though, Regeneration takes care of hero's healing and with DEF 10 and ATT 10 he does more attack than spell damage now. Great Eagle turned out great for exploring.

What units do you summon?

IIRC you get 2 more spheres when the Lizardmen are offered as a choice (regardless of the choice), and 2 when the next big racial choice occurs, for 7 total. You want 4 spheres all the same by the end, so that you can access the most advanced spells. FYI picking the lizardmen is considered a betrayal of the Keepers/Cult of Storms, so don't do it if you want to roleplay as a loyalist.

http://arek.paranoya.info/aow/index.html
list of spells here

I'd go life 4 and summon gold dragons personally.

You get 2 new spheres each time you choose a new race to lead.

I'd say 4 Air. 3 Air nets you Wind Walking (basically floating for the recipient), and 4 Air nets you Air Elementals.

After that, I'd say some combination of Life and Fire. You don't need Water because you already have some healing in Regeneration and Remedy, and the only other campaign-noteworthy spell from T1/T2 water is Water Walking. Wind Walking is better (but more expensive) than Water Walking.

1 Fire gives you Fury, which combined with Bless gives you +1 Att, +2 Dam, and +1 Resist (equivalent of 30 hero skill points). 2 Fire nets you Fire Halo, which can let you walk on lava if you run into any.

More Life gives you more mana from Life nodes, which are more abundant on a Keeper or High Men path. Holy Champion from 2 Life is good for fighting evil if you have the mana income to cast and maintain it.

I'd probably go +2 Air, +1 Life, +1 Fire for your remaining four spheres (in that order). Honestly, only Wind Walking, Liquid Form, and maybe Animate Ruins (if you raze and animate cities instead of migrating them) are the only spells higher than T2 that are noteworthy in the campaign because you simply don't have the casting ability or mana to cast the higher tier spells, especially when you've got a hasted hero running around murdering everything. Later in the campaign, I could win some maps before I could physically cast some of the T4 spells.

Air's real draw is Haste and Great Eagles.

In the campaign, I'll summon Eagles, and that's about it. Maybe Black Spiders if I don't have Eagles and it is early in the campaign. Nothing else is both relatively cheap AND more useful than having a Remedy/Healing Water on hand to heal my hero, especially when my hasted hero outpaces them.

I don't pick up higher levels of Spell Casting until around hero level 20, so I never have the casting skill for serious casting (and the campaigns don't always provide the mana to support it anyway).

In single player or multiplayer, the other low level summons do see use, depending on the map and what you have available. Eagles are still pretty much top tier choice for the non-T4 summons (and situationaly more useful than some of those summons). Frogs are wimpy, but they are useful if you don't have Eagles and you need to deal with water. Spiders are good for swarming over walls. Boars are cheap, disposable muscle to bulk out an army. I've never been in a position where I could make use of Fire Sprites, so I don't have any commentary on their usefulness. They can cross lava, but I don't generally feel threatened by Fire Sprites when I see them in combat.

Is the cavalry a special case or can you get units/heroes to attack one round earlier using say, haste? What's the boundary if so? Can a hero get two rounds of un-targetable melee like this?
Sure but then your hero isn't a spellcaster in battle. That's fine in campaign where you try to run it as quickly as you can, but in MP having a healing available can save you against the usual hordes of archers.

What about duting a single scenario?

In a single scenario, what you have at the start is what you have for the whole map.

Is there any way of displaying mission objectives?

Heroes are a special case. They don't engage in melee and can't be targeted by melee until Round 4. They only attack on Round 2 if they have some form of ranged attack. Get a ranged attack, like one of the wands or (even better) Fire Musket and they'll be attacking with near impunity for the first few rounds of combat.

Melee units will attack on round 2 if they are fast enough. The cut off is 32 or 36 movement, I believe, which is typical cavalry speed. Titans and Warlords are both T3 units; Titans are fast enough to engage on Round 2 (36 move) while Warlords (26 move) are not.

Haste might allow sufficiently fast units to start attacking on Round 2, but I'm not sure whether the game checks for haste or if it only checks the unit's actual move (for example, I'm not sure if hasted High Men infantry (30 move base) would be allowed to attack on Round 2.

That assumes that the AI was going to cast a healing spell in auto combat at all. :)

Units, like priests and paladins, are generally very quick to use their Healing ability, but I can't remember ever seeing a hero cast a healing spell in auto combat, even when there was an injured unit. Perhaps casting attack spells trumps casting healing spells; if I remember, I might test whether the AI will use a healing spell if the hero has nothing else to do.
In the bottom left is the button to bring up the menu where you can access saving and loading. There is one option ("Background", I think?) that leads to another called something like "Description". The scenario description will show along with some info like basic victory condition and hero level limit.

The max number of heroes is a little deceptive. I think it shows when heroes will never spawn and ask to be hired; you can convert heroes to get above that maximum.

Edit to add: That description might not show all victory conditions. For example, in "Hall of Heroes", you win if you either eliminate all enemies OR capture a specific town, but I think capturing the town isn't listed as a win condition.

Any luck with that? I think that'd be neat. I haven't had an opportunity to test that yet.

Another question: how are town joining you prices calculated? I seem to see that there's a part that depends on the town size and/or upgrade level, and a part that depends on the troops present as reducing the troops reduces the cost a little. I don't know if they're multiplied.

Just tried it out. I finally set up a situation where my only "combat" spell was Slow, then I bought Remedy, Healing Water, and Healing Shower from towers, injured a Spell Casting 5 hero by running through Poison Plants, and finally sent him into auto-combat.

He sat there through 4 rounds of combat before moving into melee.

I'd guess that the AI won't cast any spells that aren't marked as "Combat" during auto-combat. When I tried with High Prayer, the hero did cast High Prayer.

That is an excellent question.

The quality and quantity of defenders matters (though it might be possible to hire some of the defenders, depending on circumstances), and the size (and quality (upgrade + wall status)?) of the city matters.

I suspect a lot of the price depends on the given city's "neutral defenders" quantity and quality, which is effectively invisible. This "neutral defenders" quality are the indies that will show up if you raze the city or if it rebels and goes neutral. Theoretically, a better city will have better "neutral defenders".

As an addition to this: A hero WILL use the "Healing" ability (not a spell, but ability) if they have it.