1) What can my soldiers use unless they are Riflemen or snipers? Close or Heavy, why are you here? Get out.
Heavy is deprecated and will be removed before 2.5 is finished. Close: SMGs/shotguns before you face armour, then just the riot shotgun with saboted slugs until you can research plasma blaster, which is really deadly. Explosives: grenade launchers and maybe Rocket Launcher depending on your style and the map. You should think of them as roles, not merely categories of weapons. Each has its particular strengths and your team should be assembled with that in mind. My personal approach is to use assault as my generalists, then bring along the other skill sets to cater to specific situations. The specialists can throw grenades (smoke, frag, flashbangs) to support other troops when they're not in an ideal position to use their primary.
2) Before I was complaining how fast it was to get plamsa weapons and alien tech. But for a good month my guys were basically doing nothing. Plasma weapons hadn't shown up, I had researched the two aliens that were attacking, and only UFO scouts and fighters were running around. So yeah I had nothing to do on Normal for about half way into April to near the end of May.
Oh I thought you were talking about a gap further in. Yes, there's a bit of a lull early on, especially for experienced players who push hard on research right away. There's actually a "beginner's month" built into campaigns at normal difficulty and lower, so try hard difficulty to get right into the action quicker.
3) Yeah I know it's RNG hates me. But something is just off when an Alien can regularly hit my guys with a Plasma Blaster, an apparently Close range weapon.
IIRC, aliens have a tendency to use the snap shot and burst modes of the plasma blaster, which are more accurate than the primary ball mode. But I'd still contend it's a case of distorted perception. Our human brains tend to exaggerate the unexpected in our heads. If a shot has a 10% chance to hit, I won't take it because it seems almost worthless. But if I face 5 rounds from a 10% plasma blaster shot each alien turn, I'm likely to get hit a few times each mission. Each time it feels like a long shot, but that's just because my brain discards all the times it didn't hit.
I'm doubting the weapon upgrades. Sure there's probably plasma grenades for the launcher but nothing for the others. I'm this close to firing all soldiers that don't have good assault or sniper stats.
That'd be crazy. Explosives guys with grenade launchers almost always rack up the most kills. And I'd hate to take on some maps, like Shelter or even Oriental, without a close specialist or two.