What functions can combinator calculus expressions compute?
To get the ball rolling, and in hopes of other people giving deeper and more detailed answers on the structure of the $\lambda$-definable functions $L'\to L'$, let me cite Corollary 20.3.3 from Barendregts' The Lambda Calculus, Its Syntax and Semantics (aka "the bible").


  Corollary 20.3.3: The function $\delta:L'^2\to L'$, defined by
  $$ \delta(M, N) = \cases{\mathrm{True}\mbox{ if }M=_{\beta\eta}N\\ \mathrm{False}\mbox{ otherwise}}$$
  is not definable in the untyped $\lambda$-calculus, i.e. there is no term $D$ such that
  $$D\ M\ N =_{\beta\eta} \delta(M,N)$$
  for all $M,N\in L'$.


The proof involves considerations on Böhm trees which give a rather strong characterization of the possible "actions" of arbitrary lambda terms on normal forms. In particular, for any non-constant closed term $F$, on can find $n\in\mathbb{N}$ and $P_1,\ldots,P_n$ such that
$$ F\ x\ P_1\ldots P_n =_{\beta\eta} x\ Q_1\ldots Q_k$$

For some $k$, $Q_1,\ldots,Q_k$. This drastically constrains the possible forms of a hypothetical $D$ that implements $\delta$, showing with a little work that such a $D$ cannot exist.