Can a shooter get their own rebound on an air ball?

Answer 1

No you cannot, unless you don't move your feet. Otherwise it would be a travelling violation.

The above interpretation is incorrect for the NCAA. Below is from the NCAA rules casebook:

Traveling

A.R. 114. A1 attempts a try at Team A's basket after having completed the

dribble. The try does not touch the backboard, the ring or the

flange or any other player. A1 runs and catches the ball before it

strikes the playing court. Is this traveling?

RULING: No. When A1 recovered his or her own try, A1 could

either dribble, pass or try again. There is no team control by either

team when a try is in flight. However, when the shot clock expires

and a try by A1 or a teammate has not struck the ring or the flange, it

shall be a violation of the shot-clock rule.

(Rule 4-70.1, 4-15.4.c and 9-11.2)

Answer 2

Only if he catches it and releases it while he is in the air

Answer 3

No