Are there any open problems left about DFAs? After studying deterministic finite state automata (DFA) in undergrad, I felt they are extremely well understood. My question is whether there is something we still don't understand about them. I don't mean generalisations of DFAs but the original unmodified DFAs we study in undergrad.

This is a vague question but I hope you get the idea. I want to understand if it is fair to say that we completely understand DFAs. So I really mean questions that are inherently about DFAs, not problems artificially made to look like a problem about DFAs. Let me give an example of such a problem. Let L be the empty language if P=NP and some fixed non-regular language if P is not NP. Can L be accepted by a DFA? This question is about DFAs, but it isn't about them in spirit. I hope my point is clear and I don't get pedantic non-answers from people.

In short is it fair to say


  We essentially completely understand DFAs.


I am sorry if it turns out that this is a huge area of research that I was not aware of and I have just insulted an entire community of people.
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.