Answer to first part of question concerning a (polynomial-time) method of counting integer lattice points for the n-simplex:
Review article - Crites, A., Goff, M., Korson, M., Patrolia, L., Wolcott, L. "Counting Lattice Points in Polyhedra."
Available here with references for Barvinok's 1994 & 1999 algorithms - http://www.math.washington.edu/~thomas/teaching/m583_s2008_web/Barvinok.pdf