I'm trying to increase my skill with proofs and at least one of my bottlenecks is skill with set theory proofs. What are good largish sources of practice problems for set theory proofs (with answers if possible)?
You could try some set theoretic books:
Thomas Jech, Set Theory (3rd Millennium ed.);
Paul Halmos, Naive Set Theory;
Azriel Levy, Basic Set Theory;
Herbert Enderton, Elements of Set Theory.
There are many many more books. Each book has many questions, you can also try and prove some of the lemmas and theorem by yourself and then compare with the given proof appearing in the book.
The important thing is not to cheat yourself, if you're writing a proof and you're not 100% convinced in something that you have written - write it again. If you are not able to write it formally - write it again.
I'm not sure of the level you are interested in, but you could try Schaum's Outline of Set Theory. The entire "Schaum's" approach is to motivate concepts by working problems.
If the cost is prohibitive, you could search Google Books with phrases like "elementary set theory", "introduction to set theory", etc. With luck, you might stumble across a full version of some other text.
Halmos' Naive Set Theory contains a few problems, though some of them are a bit difficult.
I hope you will like this book also: Problems and Theorems in Classical Set Theory (Problem Books in Mathematics)
by: Péter Komjáth, Vilmos Totik,
there are various sample proof pdf online you can use to get yourself better antiquated with what style of proof to use. this might help