I'm at work, so I don't have the time to give it the response it deserves, but I'll try to answer your initial question and provide a few thoughts.

Class & Strategy

Firstly, I'm a soldier. This seemed the best choice to me, because you can take teammates with you to have access to whatever powers you want, largely negating the advantages of other classes. The soldier's powers all have very low cooldowns, so you can use them often, rather than saving them for the perfect moment. Plus, they're VERY powerful. When combined with a sniper rifle, the time slow ability (I don't know why I'm drawing a blank on its name right now) is almost too good. Walk into room, direct two teammates into cover, sit in cover myself. Watch for a guy who's about to pop his head up, hit time-slow, pop him in the head.

Even with no damage upgrades (I'm not far yet), regular troopers go down in one shot. It's worth noting that this is true for the vast majority of heavy troopers, who fire the explosive things. This is particularly important for team survivability. My guys seem fairly good at ducking into cover before they get shot to pieces, but getting caught in an explosion (which also stuns them for a bit, allowing the other dudes to shoot 'em up) tends to get them killed. Even for the guys that the sniper rifle doesn't take down in one hit though, it usually takes down all their defenses in one hit, which allows for a quick incinerate or whatever.

Generally, I just sit at range, and pick them off. Because of time being slowed I don't hardly get hit when I pop out of cover, and all I have to wait for is the recharge time on the ability, which is just a couple seconds. When a melee critter comes loping along, I just direct my mates to concentrate fire on it. They're pretty much able to handle any single enemy. When I get to a lot of husks and whatnot I'll change my playstyle some, but 95% of battles are ranged firefights, and I can't really imagine anything more effective than what I'm doing. I especially like how easy it is to take out bosses. It's pretty common for bosses to sit at the back of a room and direct their minions at you. I just follow their lead and sit at the back of MY room, and take them out in five or six shots, ignoring the minions entirely.

Teammates

I'm trying to give everyone a chance this time around. I'm mainly interested in hearing new dialog and responses and whatnot, rather than having the very most effective team possible. Given that, I've tried everyone I have (Miranda, Jacob, Mordin, Grunt, Garrus, and Jack, currently), and find that I'm most effective with Garrus and Mordin. I also like their responses best, too, which is nice. I agree that Miranda would probably be a better choice than Garrus, but I used her for my entire first playthrough, so she's benched this time around.

Weapons

I absolutely agree that the battle rifle is awesome, and I definitely use that a lot. Like I said before, though, the sniper rifle is also amazing. I usually try to get a hit on every enemy with the sniper rifle, and then pop out of cover with single battle-rifle bursts to clean up those with the gall to retain health. I think the upgrades thing you were talking about must just depend on where you go, though. My first playthrough, I had every other weapon at 3/5 or 4/5 before I found my first (maybe second) assault rifle upgrade. It was awful. heh. So yeah, I think it just depends on what areas you visit.

Anyway, all I have time for for now.