Question

Necessary and sufficient condition for diagonalizability

Answer 1

Every matrix can be put in that form, the diagonalizable ones are the ones with each Jordan block just a single entry instead of a square matrix of dimension greater than 1. That happens when the "geometric multiplicity" and "algebraic multiplicity" coincide, aka there are actually linearly independent eigenvectors for each eigenvalue. You can play around with matrices with 1s above the diagonal to see how this works out.