what is easy for minor-excluded graphs?
The most general result known is by Grohe.  A summary was presented in July 2010:


Martin Grohe, Fixed-Point Definability and Polynomial Time on Graphs with Excluded Minors, LICS 2010.  (PDF)


In short, any statement that is expressible in fixed-point logic with counting has a polynomial-time algorithm on classes of graphs with at least one excluded minor.  (FP+C is first-order logic augmented with a fixed-point operator and a predicate that gives the cardinality of definable sets of vertices).  The key idea is that excluding a minor allows the graphs in the class to have ordered treelike decompositions that are definable in fixed-point logic (without counting).

So a large class of answers to your question can be obtained by considering properties that are definable in FP+C but that are hard to count.



Edit: I'm not sure this actually answers your question, even less so for your update.  The pointer to and statement of Grohe's result are correct, but I don't think the struck out text is relevant for your question.  (Thanks to Stephan Kreutzer for pointing this out.)  It might be worth clarifying: do you want a counting problem that is difficult in general but easy on minor-excluded classes, or a decision problem?