Which of these situations are a catch?

Wikipedia defines caught as

A batsman is out caught if a fielder catches the ball fully within the field of play without it bouncing once the ball has touched the striker's bat or glove holding the bat. If a batsman could be given out caught or by any other method except bowled, 'caught' takes precedence.

Law 33 of the Laws of cricket has sub-clauses defining fair-catches, one of which is:

33.2.2.3 a fielder catches the ball after it has touched the wicket, an umpire, another fielder, a runner or the other batsman.

So, to answer your various scenarios,

If the ball hits the arm pad of the batsman and is caught by the fielder.

No. This will not be a dismissal as it has neither touched the the batsman's bat nor the glove holding the bat.

If the ball hits the leg pad first, then hits the bat and is finally caught by the fielder.

Yes. This is considered a legal catch and the batsman is dismissed.

If the ball hits the bat first, then hits the umpire and is finally caught by fielder.

Yes. Refer 33.2.2.3 above.

If the ball hits the bat first, then hits the batting partner and is finally caught by the fielder.

Yes. Refer 33.2.2.3 above.