Regarding Question #1.

There's one obvious dimensional obstruction (for example consider the Hausdorff dimension of A).

There is some research about it in the 1-dimensional case, such as the well-known theorem that K+K contains an interval, and related conjectures and works by Pallis, Furstenberg and Yoccoz. Even this one-dimensional theory is not complete as far as I know, so characterizing such a statement in larger dimensions would seem improbable now.

If say your set A is the product of two sets $A=A_{x} \times A_{y}$, such that $A_{x}+A_{x}$ contains an interval, and $A_{y}+A_{y}$ contains an interval, then one can have that the sumset of the product $A$ will contain a rectangle.

The one thing that can help you tackle the problem in larger dimensions, is the fact that you can sometimes say something smart about the projection of your set in a.e. direction (especially if your set is self-similar).
For example this is the content of early works by Furstenberg (back in the 60s), and even some recent works (such as Hochman-Shmerkin - arXiv:0910.1956).

I hope that by taking two independent generic directions, you can say something smart about the projections (Hausdorff dimension? entropy estimates?), and then maybe one can use the one-dimensional theory to get that the sumset of the projection contains an interval. Of-course, it is not enough to get an interval in the projection, because your set might not be rectangular set itself, but maybe if your set if self-similiar, you can wiggle the pieces around to construct an inner rectangle.

By Masterand's theorem, I'm guessing that a resonable bet here would be $dim_{H}(A)>1$ (or maybe even take the upper packing dimension of $A$ to be bigger than $1$) for "nice enoguh" sets $A$, although I'm pretty sure this is open in general (because of Pallis' conjecture).