My actual questions are at the bottom, so feel free to skip ahead if you don't care about my backstory.
I'm in the mid-to-late stages of UFO: Aftermath, and it ain't goin' well. Everything was going fairly smoothly, at first; no transgenant was a match for the survivor-warriors I'd trained and equipped, and for the greys, while tough, could still be taken down with good tactics and the right weapons. The only trouble is, I think that the game has grown ahead of my characters.
It started a little after I finished the Area 51 mission. I started encountering larger numbers of aliens with better weapons, including some nasty warp-based mind-control beams. Initially, I could still take these on, mostly because by then everyone had high-powered rifles and some laser weapons, but then I started encountering aliens in Sky Armor and Bio Armor, with rocket launchers and other weapons that could kill my most heavily armored guys in 2-3 shots. These, too, I managed to (barely) beat, via liberal application of rocket and grenade launchers. But at the same time, the Biomass appeared, and a huge clump of it turned up right in the middle of North America, where about half my bases were. Before I could get the biomass repulsors up, about a quarter of my military-industrial base got eaten.
The war started going downhill. Miraculously, I managed to find some heavy machine guns and collapsible weapons in the Stalingrad military base, so fighting the aliens who were equipped with Bio armor and superweapons was manageable, but I had to save and reload several times to get through the Project Dreamland mission without getting anyone killed, and almost all of the guys I sent in had to be hospitalized anyway.
Now EVERY battle I fight is against these ungodly strong aliens who tear through my armor like tissue paper and always outnumber me. Any engagement is largely determined by who shoots first, and almost the entire squad is in the hospital right now. Half of my bases are converted to biomass repulsors, and the remaining half (mostly located in Europe) are in danger of encroaching biomass from Africa in the south and Asia in the east. Oddly, South America is largely untouched, but for some reason I never get the opportunity to build new bases there. I honestly don't know now if I have the resources to win the war.
Long story short, after all that I've seen and done, I think it'd be a good idea to start over, because I think I'm losing. Before I do that, though, I have some questions I need answered.
1. First and foremost: are the key "events" in this game (increasing grades of transgenics/aliens, the coming of the biomass, UFO attacks on bases) completely random, run on a time trigger, or running on an event trigger?
I figure that if there's an event trigger to these things, then I can exploit it: I can purposefully hold off on doing plot missions like Area 51 until I've conquered multiple continents and have a strong industrial complex to back me up. However, if there's a time trigger, then I'm obliged to race around doing as much as I can in as little time as possible. If it's random, I might be able to cheat my way through with multiple savegames to fall back on if events start looking unfavorable.
2. What can I do to guarantee that I get better technology sooner?
I don't EVER want to have weapons or armor that are very inferior to the enemy's. I know now that some technologies don't appear until their prerequisites are discovered, and how the research system works, but I would like to know if there's any tricks to being able to get things like Enhanced Heavy Armor early, because I desperately need it.
3. What did I do wrong?
I have no idea if I was just unlucky or if I made some serious tactical blunders in my overall strategy. I tried to research and develop everything I could as soon as it became available, with priority given to things that would make the war effort easier. I tried to seize bases first and do other types of missions second. I developed my soldiers with the specific intent of training them to be "Excellent" in every way. Then the Biomass just fucked everything up. It's preventing me from expanding any further, and I don't know how it got started growing right in the middle of the USA, and without better armor and/or weapons I don't know if I can keep going with the plot. I'd like to know what, if anything, I could have done better.

In previous X-Com games, the aliens technology would increase at the end of every month or week for X-COM3 depending on the previous month's score, although they'd still increase anyway. I suspect a similar system is in place in these UFO games. As time progresses and you win missions, the difficulty adjusts so you are ALWAYS chasing the alien's level of technology. This keeps the challenge relatively high throughout, until you reach the end game plateau where you're up against the same type of ridiculously over-powered Reticulans most missions.
With Aftermath, there is some element of luck involved in getting certain weapons sooner when you capture bases, or at least the ammo needed to run them (ie you might find a really decent sniper gun early, but then not find any ammo for it for another 4 months!). Really you don't need much more than a couple of collapsable MGs to win most missions after a time. I'd ignore the Laser weapons as well. They are accurate but don't do enough damage to most foes. Best bet is to use Collapsable guns and your own Enhanced Plasma guns. They chew through everything quickly as the key to winning firefights in Aftermath is to get the most amount of firepower centred on one spot as fast as possible. Tactics? Few required. Also make sure you specialise a few troops in their chosen weapons field - get them to heroic and super heroic levels and don't skimp on the aliens skill. You'll be able to pull off longer range burst shots that way, and a high Alien skill rating massively increases the damage you do.
As to the biomass, you need to research a way to destroy the biomass nodes. If you can't destroy the nodes, the biomass can only be repulsed but never beaten. It can be tricky waiting an opportunity to get a mission that enables you to gather the biomass for research if I recall. It's important to target the UFOs that plant new biomass nodes. If you don't stop them, the biomass will pop up elsewhere too.
Finally I'm not sure you can put off the special missions - I think they occur just once and if you fail to do them, you'll be unable to win the game. I'm not 100% sure on this but I think that's the case. It's been a few years since I got to that stage of the game so my memory is a little rusty and I usually stop playing about half-way through the game whenever I've played it since. Once the Reticulans pop up with their super fun destroying guns, the game becomes a chore.

Bah. I HATE it when so many variables are left to chance... but fine. I think I know what to do now. Thank you for your help.

I have to admit that one of my bigger problems with Aftermath is that it's idea of increasing the difficulty is giving the aliens all rocket launchers and mortars and having them head straight for you.
I could just barely manage anything except base defense missions, and the final mission was also a bit hard.
For base defense, I eventually just learned that it's better to start chucking a river of grenades as soon as you see a bad guy.
And for the final mission, my advice is take along a couple of those psi projectors to un-control captured men, 2 or 3 rocket launchers, and maybe some warp weaponry. And keep a save from before starting the mission. You'll find out why in the last room...

For the sake of experimentation, I have started my game over and have purposefully avoided taking Area 51 for some time. I conquered most of North and South America, and then just let the clock run without doing any missions. All I did was take the time to shoot down any UFOs that came by.
I finally got a game over on April 20th or so, and in all that time I didn't see a single speck of biomass anywhere on the planet. However, I also didn't get the opportunity to research better tech, or get any missions related to discovering new stuff.
I hypothesize that this game works on both a timer and an event-based system. If you don't do missions (plot-related or otherwise), the game's difficulty doesn't increase, but you also don't develop your power at all. It may be that never doing plot missions WOULD enable me to slowly conquer the whole world, but then I wouldn't have any of the really good weapons and wouldn't be able to actually finish the game.
I think this is actually pretty good news, because it means that I DON'T need to desperately scramble to get plot missions done, but at the same time it's apparent that I'll have to do them sooner rather than later if I want to increase my power.
That said, I have a question about collapsible weapons:
Whenever any soldier I have uses one, he takes a few seconds setting it up before firing, and immediately dismantles it when the target is killed. Is there some way that I can make my guys set up their weapons in advance and/or make them stay up in a fixed place? This is important mostly because in the time it takes for a dude to unload his weapon, the aliens with hand-held warp guns can tear him up.

It's been ages since I've used them, but you can set them up and wait for the enemy to appear. I'm not sure if you have to use the CROUCH function. There is definitely a way, even if it isn't that. In UFO missions and Alien base missions, setting up collapsable MGs and railguns to cover the corridors the aliens pour into from the starting position is a must to do straight away. Don't move those guns until there are no more nearby threats. They take an age to set up and dismantle but once they are in position they are the most deadly weapons in the game. The MG variant seems to be the best. Although it only fires 10 round bursts it usually does some damage from a long range and with a decent set of stats, it should kill just about anything in 2-3 bursts. It also sounds cool and has 60 bursts which should be sufficient for any mission you'll go on... even the last.

You're right, it was the CROUCH button. They are definitely amazing guns... I have half the squad packing them now, and it's amazing to see them tear apart the most heavily armored enemies from across the map. Beautiful...