("thinking outside the box"...) this is a somewhat contrived problem involving DFAs (have not seen it studied elsewhere) but manifests a theme in TCS that even many apparently "simple" computational objects (like DFAs) can have complex properties, also an aspect/theme embodied in Rices theorem. (in some ways the ultimate "complexity" is "undecidability" aka Turing completeness.)
consider a family of DFAs and an "exponentiation" operator "↑". this operator takes a regular expression (RE) "x↑n" and repeats it $n$ times. ie for any RE $x$ and finite $n$, $x↑n$ is a RE (and therefore also a DFA). this operator is considered in various contexts and in some important problems eg one of the first problems proven EXPSPACE-complete.
now consider a family of DFAs $DFA_n$ which is built from a single RL expression $DFA'$ (not exactly/strictly a DFA) with embedded instances of the exponentiation operator in the form "↑n" where here "↑n" is a symbol (but otherwise $DFA'$ is a RL/DFA). then for every finite $n$, $DFA_n$, constructed by replacing/substituting the instances of "x↑n" in $DFA'$ by n repeated instances of x, is also a RL (and a DFA).
(now as usual let $\Sigma$ be the language symbol set/alphabet.) claim:
the problem "does there exist an $n$ such that $DFA_n ≠ \Sigma*$" is undecidable.
the claim follows from constructions in  where basically a regular expression is constructed using exponentiation that simulates a TM computation on a specific input iff it is not equal to the "full language" $\Sigma*$. the exponentiation is used to reflect the maximum tape width used in the TM computational tableau accepting the word in the language. this construction can also be found in . a halting computation can be found iff there is a finite $n$ tape width for that computational tableau accepting the word.
now, to tie this in more with the question, while this is not widely noted (considered trivial by some), many open problems in TCS/mathematics are tightly connected with undecidability in that given an oracle for the halting problem, they can be "solved".
therefore, in a sense, tying this all together using this basic problem about DFAs that is undecidable, there will always be open problems about DFAs, because there will always be "open" problems about DFAs (such as this one) equivalent to undecidable problems. in fact using Rices theorem in reverse as this construction does in some ways, basically any relatively "simple" yet nontrivial computational property in TCS can be used to construct undecidable problems.
 Word problems requiring exponential time / Stockmeyer & Meyer
 Meyer, A.R. and L. Stockmeyer. The equivalence problem for regular expressions with squaring requires exponential space. 13th IEEE Symposium on Switching and Automata Theory, Oct 1972, pp.125–129.
 Introduction to languages, automata and computation / Hopcroft/Ullman.