player A shoots the ball and it touches the rim and at the same moment he/she goes out of bounds and is the first player to touch the ball after he is established back on the floor. Can he/she do that?

Perfectly legal.If you have it,Casebook play 7.1.1SitB covers this play almost verbatim-"if the try is unsuccessful,A1 may come back onto the court and regain control since A1 did not leave the court voluntarily and did not have control of the ball when he/she did.".

Welcome to the board,Jeffrey.

And, of course, only one foot need touch in bounds, so long as the other foot is no longer touching OOB.