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 http://cstheory.stackexchange.com/ with a more obviously computer-centric version of your question.