The keys for fast composting are chopping the material as small as possible, keeping the pile damp but not wet at all times, a proper ratio of browns to greens, and daily turning. As long as those parts are all in place, you 'll get great compost as quickly as possible. If any of them are wrong, it slows the process.
Size can also play a part in heating up of a pile. I have one of those storebought black bin composters, as well as a large homemade bin. The storebought bin still gets pretty darn warm and works pretty well, but isn't capable of getting as hot as my homemade one due to lack of volume.
Just the standard woodpallets and boards in a cube set-up. I threw some landscape fabric I pulled out of the yard on them to help keep materials in and still allow airflow. Nothing fancy, but it's been working pretty well for me. I keep an old tarp over the top right now to keep moisture in, otherwise the top several inches dries out quickly in the summer heat.
Not right now, no. I'm planning on making some adjustments on it this Fall but right now I've been using the heat and moisture from it to kill/rot some laurel stumps. It's worked well elsewhere, but my present location is rather awkward, on a steep hill. I have one side more open than the other 3, so my turning process right now is to scoop about half the material into trash cans and then turn the center to the outside.
That is one huge plus of the storebought one: it may be small, but it takes less than 10 minutes to turn everything.
It varies on the time of year: ground up ramial branches with leaves attached from aforementioned laurels, weeds and grass clippings, fallen maple leaves in Fall/Winter. Also, from late Fall until Spring lots of kelp, and in summer other types of seaweed. Sometimes I grab young podless Scotch broom at the beach as well as that's the only place I can be sure it hasn't been sprayed (lots of work goes into hand removing it by volunteer groups, but of course it always comes back.)
Oh... I mostly have tree leaves but haven't tried to compost yet, I've been making do with comfrey tea from about 30 plants I started from 2 crowns. Ha! I got stung today disturbing a bees nest while picking comfrey leaves. The wad of leaves made a pretty good fan to beat them off as I ran down the road, lol.