What can I say about the Parikh map of a CSL? Write $\Psi$ as the Parikh map--i.e., $\Psi(w) = \{(\#_\sigma(w))_{\sigma\in \Sigma}\vert w\in L\}$, where $\#_\sigma(w)$ is the number of times $\sigma$ appears in $w$. It's well-known that, for a CFL $L$, $\Psi(L)$ is a semilinear set (this is Parikh's theorem). Some other interesting things are known, but I have found nothing about the Parikh map of a context-sensitive language. In particular, 


  what can I say about $\Psi(L_2 - L_1)$
  or $\Psi(\bar{L}_1)$ if $L_1, L_2$ are
  context-free? For instance, if I let
  $\phi(L) = \{\sum_\sigma \#_\sigma(w)\vert w\in L\} = \{|w|
 \vert w\in L\}$, is it possible that
  there is a CFL $L$ such that
  $\phi(\bar{L}) = \{n!\vert n\in
 \mathbb{N}\}$? (or any other
  'increasing' sequence converging in
  $\hat{\mathbb{Z}}$, for that matter.)
Regarding the second part of your question: If you choose your CFL $L$ to be the set of all invalid computations of a Turing machine $M$ (see, e.g., Chapter 8.6 in the first edition of "Introduction to Automata Theory, Languages, and Computation"), $\phi(\overline{L})$ is the set of all lengths of encodings of accepting computations of $M$.

Although this does not directly answer your question, you can use this approach to construct quite complicated sets.