Decidability of the language of DFAs accepting only odd-length strings
Every DFA can be simplified by a Turing Machine, so you can simplify the DFA you receive as input and compare it to the simplified DFA that accepts odd-length strings.

On input $M$:


Simplify $M$. Remove every unreachable state and transition from $M$ (disjoint from start state)
If the start state of $M$ is reject, and the only other state in $M$ is accept, and for every $w \in \Gamma, \exists$ a $w-arrow$ from accept to reject and from reject to accept, then accept. Otherwise reject.