Are there any books containing collections of automata problems? You can find specialized books consisting entirely or problems from particular math domains (e.g. linear algebra, polynomials, combinatorics), but I've yet to find such a book for automata of any kind.
Sort of indirectly, this book is about research level automata problems:

http://www.amazon.ca/Automatic-Sequences-Theory-Applications-Generalizations/dp/0521823323

An automatic sequence is essentially a set of strings which can be computed by a DFA.