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Your problem is NP-complete, by reduction from 3SAT. Let $\varphi$ be a 3SAT formula with $m$ clauses and on the $n$ variables $x_1,\dots,x_n$. Construct a NFA over the alphabet $\Sigma=\{0,1\}$ as follows. The $n$-bit input to the NFA is treated as an assignment to $x_1,\dots,x_n$. The NFA first nondeterministically selects one of the $m$ clauses. Then,...


Yes, even in the purely existential case. See

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