All Questions

I will start the story by the end: Is there some characterization of (some of) the singularities arising from the Minimal Model Program (canonical, terminal, log-...) in terms of commutative algebra ?...

Let X be an irreducible deminormal variety such that the normalisation is Gorenstein. Does it follow that X is also Gorenstein? for deminormal definition, see https://arxiv.org/pdf/1506.02002.pdf