Is finding the minimum regular expression an NP-complete problem?
Yes, it is NP-Hard.  Pitt and Warmuth showed that finding the smallest DFA consistent with a given sample cannot be approximated to within $OPT^k$ for any constant $k$, unless $P = NP$.

Regarding the learning question: Kearns and Valiant proved that you can encode RSA into a DFA.  So, even if the labeled examples come from the uniform distribution, being able to generalize to future examples (also even coming from the uniform distribution) would break RSA.  Hence, we think that in the worst case, having labeled examples does not help with learning a DFA (in the PAC model).  This is one of the classic cryptographic hardness results for learning.

Both of these issues are intertwined due to what we call the Occam's Razor Theorem.  It basically states that if we have a procedure for finding the smallest hypothesis from a given class that's consistent with a sample labeled by a hypothesis from the same class, then we can PAC learn that class.  So, given the RSA hardness result, we would expect that finding the smallest consistent DFA would be hard in general!

To add a positive learning result, Angluin showed that you can learn a DFA if you get to make up your own examples, but it requires the additional power being able to ask "is my current hypothesis correct?"  This was also a seminal paper in learning.

To answer your other question, this is all indeed related to Kolmogorov complexity, as the learning problem becomes easier when the canonical representation of the target DFA has low complexity.