If a player for team A loses his dribble in the back court under the opponents basket and all other 4 players from team A have run off to the front court, along with the 5 players from team B, which obviously he has no teammates to pass to at the moment.

Can he shoot the ball towards Team B's goal in the back court and get his own rebound to start another dribble?

Remember, if team A makes basket into team B's basket, it scores 2 points for team B.

No, this would be a double dribble.

Yup
 

Throwing the ball off of the opponents' backboard constitutes a dribble. Even if he throws it off the board first, then catches it, then starts to dribble, it's a DD. I always wondered why that was in the rule book, but I suppose your scenario answers that question.

How about if he hits the rim?

Re: Yup
 

He's already dribbled. Once he throws the ball off the backboard and catches it, he's violated. He doesn't even get an opportunity to dribble again. The violation has already occurred.

That doesn't constitute a dribble. However, it doesn't give him another dribble either. It's not a shot, so there's no loss of team control. If he runs and gets the ball, he's traveled, the same as if he threw himself a pass.