Regarding Pfaffian Methods in Counting and Combinatorics
(This is an interesting question for me because I'm also reading about the Pfaffian.) 

I suggest the following references:


Chapter 8 of the awesome book Matching Theory by Lovasz and Plummer.
The paper Division-Free Algorithms for the Determinant and the Pfaﬃan: Algebraic and Combinatorial Approaches by Rote. This paper relates the Pfaffian with the combinatorial object called alternating closed walk sequence (aka. alternating clow sequence), which gives us a dynamic (division-free) algorithm for computing the Pfaffian.
The paper Nonintersecting paths, pfaﬃans, and plane partitions by Stembridge, which shows how to use the Pfaffian to enumerate configurations in combinatorics. The paper also gives combinatorial proofs of basic identities, e.g.,


  If $A$ is a skew-symmetric matrix, then $\mathsf{pf}(A)^2 = \mathsf{det}(A)$.