Determining if a word of specific length exists that is not accepted by a NFA
Your problem is NP-hard, 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, it checks whether the input satisfies that clause, and if so, rejects; otherwise accepts.  Note that if the input $x$ satisfies $\varphi$, then the NFA rejects; otherwise, the NFA accepts.

Now there exists an input to the NFA of length $n$ that causes the NFA to reject, if and only if there is a satisfying assignment for $\varphi$.  That means that your problem is at least as hard as 3SAT.