Hi there, I think we had better give you a start here...
You have combined together a few ideas that come from very different areas of inquiry.
In one direction, kissing numbers and Minkowski-Hlawka (Milnor and Husemoller, page 31) see Table 1.3 on pages 15-17 of SPLAG, that is Sphere Packings, Lattices and Groups by Conway and Sloane. Self-dual is typically called "unimodular," see the bottom of page 53. Easier introductions that will still lie in comfortable territory are Lattices and Codes by Wolfgang Ebeling, also From Error Correcting Codes Through Sphere Packings to Simple Groups by Thomas M. Thompson. Note that your Thompson is J. G. Thompson.
See http://en.wikipedia.org/wiki/Unimodular_lattice
As to exact numbers of representations, I actually recommend a much earlier book, The Arithmetic Theory of Quadratic Forms by Burton W. Jones. I see that in Rational Quadratic Forms by Cassels, he does three squares on page 150, Lemma 6.4, then four squares on page 152, Lemma 6.5.
In a few famous cases, notably four squares and eight squares, the exact number of representations has a fairly clean expression, due to Jacobi. http://en.wikipedia.org/wiki/Jacobi%27s_four-square_theorem
See http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/unimodular.html
Finally, you might try https://cstheory.stackexchange.com/ with a more obviously computer-centric version of your question.
