Let $A$ be a finite set in $\mathbb{R}^2$ of $k^2$ elements and consider a set $B=\{x_1,x_2,x_3,x_4\}$ such that the points in $B$ are in general position (no three points on a line).

Question 1: Is it true that $|A+B|\geq (k+1)^2$?

Question 2: Is it true that $|\cap_{i}(A+x_i)|\leq (k-1)^2$?

Question 3: If either of 1-2 are true, is it the case that one of the extremal sets $A$ is a discrete square?