State complexity of difference of two languages
This is correct, and moreover it is tight for $n = 1$, even for a binary alphabet (but not for a unary alphabet!). Indeed, it is well-known that there are language accepted by $m$-state NFAs which require $2^m$ states for a DFA. Since the DFA complexity of a language and its complement coincide, taking $A_1 = \Sigma^*$ we obtain a tight example with $n = 1$ and any $m$.
It is an interesting question whether the bound is tight for all $n$. This is likely the case if we take for $L(A_1)$ the language $(\Sigma^n)^*$.