Does diagonalization captures the essence of class separation ? I don't remember having seen a class separation not based on diagonalization and relativization results.  Diagonalization could still be used to separate remaining known classes, because non-relativizing arguments might still be used in the diagonalization conclusion, or in the diagonalized Turing machine construction.  Here are some related questions:


  Are there class separation proofs not based on diagonalization?


And if so


  Can we find a self-reference mechanism behind them ?


Further,


  does every class separation have a "canonical natural" proof (in an informal sense)?


If so, we should try to find non-relativizing arguments, rather than other proof schemes for open questions.


  Can every non-diagonal proof be rewritten into a diagonal one ?
Depends on how you formalize diagonalization. Kozen has a paper that shows any complexity class separation must be a diagonalization proof.