1) The check engine thing is common, you need to jumper the diagnostic terminals (they'll be under a cap labeled "diagnosis" in the engine compartment. There are plenty of threads on the forum about how to check them, also how to download the Big Green Book--probably the best resource for working on MR2s. If it's going off and on it's probably the O2 sensor, but get the codes first--no point replacing something if that's not the problem!
2) you're probably right about the switch, I'd take it out, clean it with electrical connector cleaner and put it back together.
3) possibly a tire, possibly something worn/bent/broken in your suspension (those bushings don't last forever, especially where it's hot and dry!) or possibly your alignment is bad. Since you're new to driving (and presumably mechanics), I'd have to advise you to take it to an alignment shop, they'll tell you what (if anything) is worn or bent, and adjust things so your wheels point straight down the road. It's probably $75 well spent, even if they say you need to replace some other suspension parts.
5) the 5th gear popout is common to most all mk1 mr2s, replacing your transmission with another used transmission probably won't fix it. It's a design "feature," if you hold your hand on the shift lever it'll stay in gear. You'll notice that when you take your foot off the gas and then accelerate, the shift lever pushes back. This is partly design, and partly because the engine/trans shifts on the motor mounts. Kirk's motor mount inserts (available at
Twos R Us - Parts for Toyotas from Club MR2) are supposed to help. I've got some but haven't installed yet so I can't give an endorsement yet.