(I am especially interested in abelian surfaces and characteristic 0).
How bad is the moduli stack of abelian varieties (with no polarization or level structure)? Is it an Artin stack? DM (Deligne-Mumford) stack?
How bad is the is the stack of abelian varieties with full 2 level structure (so with a basis for $A$)?
Consider the maps from either 2) above or stack of principally polarized abelian varieties to 1) above. Are these maps smooth, are the geometric fibers finite (ie, are there only finitely many principal polarizations on an abelian variety)?
Neither moduli space is a stack because every point has the automorphism $-1$ but the same is true for the moduli stack of elliptic curves and that is still a DM stack and not too bad.
Even in characteristic 0, the CM locus is higher dimensional so the "especially stacky" locus has high dimension but I don't know how serious the problem is.
For the second question, while $-1$ fixes the two level structure, I suppose a generic CM automorphism doesn't fix it so perhaps the second stack is very nice, or atleast almost as nice as that of elliptic curves?