With regards to your Q2, in quantum field theory there is a commonly used generalization of the Kirchhoff (first Symanzik) polynomial to 2-trees (2-component spanning forests). It's normally called the 2nd Symanzik polynomial. I'm not sure if it can generalize to k-spanning forests.
To calculated the 2nd Symanzik polynomial you need to associate a variable with each vertex. (In QFT this is the incoming momentum at that vertex.)
There's a nice recent article which discusses some of this
Feynman graph polynomials (arXiv:1002.3458v3)
I also made a Mathematica demonstration that lets you draw graphs and calculates the polynomials.
Scalar Feynman Diagrams And Symanzik Polynomials.
The classic reference is
N. Nakanishi, Graph Theory and Feynman Integrals, Newark, NJ: Gordon and Breach, 1971.