Lower bounds for Polynomials computing the boolean functions
Regarding Question C: Number of monomials is essentially the same thing as circuit size of depth-2 algebraic circuits (unless the polynomial is a product of linear polynomials, in which case one could also consider a depth-2 circuits with a multiplication gate at the top). 

Degree is very closely related to the depth of algebraic circuits, as in the results of Hyafil and Valiant-Skyum-Berkowitz-Rackoff.