Can a Turing Machine decide if a regular expression matches all strings of exactly some length $l \geq 1$?
I think this procedure should work...

First, construct a DFA for the language. Now, start tracing all possible paths of length $k$ (start with $k = 1$). Call $S_k$ the set of states reached by strings of length $k$.

If at any point you find $S_k \subseteq A$, the set of accepting states, the answer is yes, and return.

If, on the other hand, you find that $S_{k} = S_{k'}$, where $k' &lt; k$, before you find an $S_k$ that works, you know that you'll never find an $S_k$ that works, and you can stop looking. The reason is that the next length you try, $k + 1$, will yield $S_{k+1} = S_{k'+1}$, another set that didn't work... and so on.

Note that this procedure is guaranteed to terminate since there are a finite number of possible $S_k$; since they're subsets of the set of all states, $S$, there are no more than $2^{|S|}$ of them. By the pigeonhole principle, if you've tried enough candidates, you'll either have already found one that worked, or you'll have tried some set more than once.

Apologies in advance if this is completely missing the point of the question, blatantly wrong or intellectually hilarious.