Q: What are the ways a defender push the opponent out of the key legally and is it okay to use the lower body to push the offence out?

A; In all of the above cases, it is not legal to push an opponent. Here is the official Article from the current FIBA Rulebook:

33.15. Pushing
Pushing is illegal personal contact with any part of the body where a player forcibly moves or attempts to move an opponent with or without the ball.
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