Ball tossed off own backboard (not a shot)
 

NFHS Case Play 9.5 clearly spells out that a pass off the opponents backboard is a dribble.

It also spells out that it is NOT a dribble to toss it off your own backboard.

However, it doesn't specify what tossing the ball off your own backboard (when it is NOT a try) does once a player has ended a dribble.

For instance, is it like bouncing it off another player and you are free and clear to dribble again?

Or is it "nothing" (as if it never happened) so that a player would commit a double-dribble violation if he/she bounced the ball to the floor again?

well if I'm reading it correctly then tossing off the backboard a shot or not means you get a new dribble. I've played in games where I've purposefully thrown the ball against the side just to get a new dribble.

DRIBBLE RULE
9.5 SITUATION: A1 dribbles and comes to a stop after which he/she throws
the ball against: (a) his/her own backboard; (b) the opponent’s backboard; or (c)
an official and catches the ball after each. RULING: Legal in (a); a team’s own
backboard is considered part of that team’s “equipment” and may be used. In (b)
and (c), A1 has violated; throwing the ball against an opponent’s backboard or an
official constitutes another dribble, provided A1 is first to touch the ball after it
strikes the official or the board. (4-4-5; 4-15-2; Fundamental 19)

The case play that Nevada brings up is the one that the OP brings us but it does not address whether or not that player can dribble again as the OP asked.

Most officials consider this an attempt regardless if you think it was a "non shot" or not. This would give the player the right to dribble.