Am I required to shuffle my opponent's deck?
Competitive and Professional REL

You are always required to shuffle your opponents' decks. From Section 3.9 of the Tournament Rules:


  At Competitive and Professional REL tournaments, players are required to shuffle their opponents’ decks after their owners have shuffled them


Regular REL

You are usually not obligated to shuffle your opponents' decks. However, the Head Judge may optionally require it. The very next sentence of the aforementioned section in the Tournament Rules:


  The Head Judge can require this at Regular REL tournaments as well.


Casual (No REL)

You are never obligated to shuffle your opponents' decks.

Is cutting acceptable?

The rules do not care whether or not you cut the deck. You can cut if you want, but you don't have to.

You are always obligated to give your opponent the opportunity to shuffle your deck, even if they elect not to. If you are physically unable to shuffle, you can ask a judge to shuffle for you.