Deciding if a finite automata accepts strings of any length
Let $L$ the language accepted by your DFA. The question amounts to ask whether $\{ |u| \mid u \in L \} = \mathbb{N}$. To check this, just identify all letters to a single letter, say $a$, in your DFA. You will get a NFA on the alphabet $\{a\}$. It now remains to check whether this NFA accepts $a^*$.