Are there any open problems left about DFAs?
Here is one problem described in the book "A second course in formal languages and automata theory" by Shallit.


  Let $u$ and $v$ be two distinct words with $|u|=|v|=n$.  What is
  the size of the smallest DFA that accepts $u$ but rejects $v$, or vice versa? 


Robson, in his paper "Separating strings with small automata" in 1989 proved an upper bound $O(n^{2/5}(\log n)^{3/5})$.  The best known lower bound in $\Omega(\log n)$.

For a survey see this.