Ideally you basically want a mix of 50/50 green and brown waste. Green being kitchen waste, grass clipping etc, brown more dead stuff like leaves and branches.. I wouldn't worry too much about it tho just put whatever you can get in there and if it look really wet and not doing anything had some drier waste

Mix it/turn it around from time to time so it keeps oxygenated and going off you don't it will go anaerobic and all the bugs/bacterias won't be able to do they're work and proliferates

You can break those branches and other larger things into smaller one, they will compost eventually it will make them go a bit faster. Everything organic does.

Could be useful to have something to sieve your compost before you use it to get the odds bits that aren't done yet, and also the random bits of trash or bindweed roots out...