02 Corolla with a couple codes and issues

[Krono353]

My 02 Corolla LE has codes P0115 (engine coolant temp sensor) and P0420 (catalyst below threshold). I've read a few tutorials and videos on the temp sensor, seems simple enough. The catalyst code, I've read, can be fixed by replacing the o2 sensor post-cat because usually that is the cause, not the converter (unless it's burning oil(?)). Any other thoughts? I can't afford to pay a bunch of money to maybe have it fixed, I'm going to have to do a lot of repairs myself so I'm hoping someone can give me a direction to move in.

EDIT: Also, the engines idle gets weird and goes from 900rpm to 1400+rpm. Other times I'll start the car and it'll rev past 2100rpms and drop to about 1700, then rev and drop 1 or 2 more times. If I'm moving while it's doing this I can leave my foot off the gas and the idle will pull the car while it's in gear.

Other issue: When the temp is cold outside, the car slides in gear easily (it's manual transmission btw) but when it's hot outside or the when the engine heats up it feels like it shudders, makes the whole car shake sometimes. Don't know if it's a clutch or transmission issue.

Looking at it on paper it looks like it has more issues than I thought. Please, any questions or helpful comments will be appreciated.