What's a good advanced textbook/resource for studying the complexity of counting and combinatorics?
If I understood correctly, you're looking for a book (chapter) on counting complexity. In this case, I'd recommend the book of Sanjeev Arora and Boaz Barak, which has a chapter on counting complexity. A draft of it can be found here.

Concerning Extremal Combinatorics, Wiki suggests reading Extremal and Probabilistic Combinatorics by Noga Alon and Michael Krivelevich which contains pointers to some resources (mainly graph-related).