Yes.
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Yes. It is completely possible to catch Mewtwo with an ultra ball.
No. They have the same catch rate as all the other legendary Pokemon. However, they are all in caves, so you can use a dusk ball which has twice the catch rate as an ultra ball (only in caves or at night) and is 200 poke-currency cheaper than an ultra ball. You could still get lucky with an ultra ball, but if I was using ultra balls, I would bring at least 40 of them, a Pokemon with hypnosis (or some equivalent), maybe false swipe, and save before you try! You could probably be fine with just 30 dusk balls.
Yes, the Ultra Ball has a 4x catch rate than a Premier Ball, which has the same catch rate as a normal Poke Ball. It is commonly used to catch rare or Legendary Pokemon.
Yyes!!!!!!!!!!!!!!!!!!!!!!!!!!
the ultra ball is what i used to catch it...but you could use the master ball after you get it