Relationship between context-free/decidable languages and NP
Basic Facts:


CYK-algorithm parses inputs of length $n$ for any CFG in Chomsky normal form in time in $\mathcal{O}(n^3)$
Every CFL is generated by a CFG in Chomsky normal form
Every regular language is (also) context-free.


That answers immediately all those questions:


  What is the relationship between NP and context-free/regular languages? Are there context-free languages in NP? Are there regular languages in NP?


$\{a^nb^nc^n \mid n \geq 0\}$ is not context-free (by pumping lemma). It can obviously be decided by a (D)TM in quadratic time. That answers:


  Are there languages in NP that are not
  context-free? Are there languages in
  NP that are not regular?


And as for


  Are there also decision problems that
  are decidable but not in NP?


Yes, see Problems that cannot be verified in polynomial time