NO, no please do not waste your money and time to paint the barrel because the paint will just peel off after awhile.
That's awesome! If it rots, put it in there. Too many things to list. The things I avoid are thorns, weeds with seeds, bones, sticks bigger than a finger, anything with urishiol, meat, dairy, anything very salty or greasy.
Just wanted to update this posting. I had to make some adjustments because it just wasn't working right (or should I say I wasn't working it right). The barrel was getting so heavy the wheels would sink into it making it nearly impossible to turn. So... we added another set of wheels. That took care of that problem. I also painted the barrel black to help it heat up in there. The stuff was staying way too wet, so I removed one of the openings on the side and drilled a bunch of small holes in various places. (I think I need to make them a bit bigger, but didn't want to ruin the barrel by making them too big.) Tumbling wasn't really mixing things up, it was just flopping over this "glob". So I've been stirring and breaking everything up with my little hand-rake. I spread out some grass clippings and let them dry up really well and mixed that in. (I'm going to try to keep some of that on hand to cover up newly added kitchen scraps.) I've been leaving the trap door open lately too. (I'm going to close it tonight since we've got good chances for rain this weekend.) It's now drying out pretty good and lots easier to stir/mix and the tumbling is working better to mix it up. Now... I just need some more goodies to add to it. Next week, I'm going to ask all my co-workers to start saving their coffee grounds for me. And I think I need to hop the fence and get me a few buckets of cow manure. I think I'm on the right track now.