Kicking Opponent or Puck

The puck is kicked by an attacking player, deflects off his stick and enters his opponent’s goal. Should the goal be allowed?

No. Rule Reference 627(c).

Had the player intentionally directed the puck into the goal with his stick, the goal would have been allowed.

See also 617(C)2 - A goal shall not be allowed if the following occurs:
The puck has been thrown or deliberately directed into the goal by any means other than a stick, even if subsequently deflecting off any player, including goalkeeper, prior to entering the goal.

Off-Sides

A player has both skates completely in the Attacking Zone the instant the puck completely crosses the blue line. Is he offside?

Yes. Rule Reference 630(a).

The position of the player’s skates at the instant the puck enters the Attacking Zone determines off-sides.

Electronic Devices

Team A is using a light on the players bench for the purpose of informing players on the ice that it is time for a line change. Is this permissible?

No. Rule Reference 308(a).

While this is not considered as an electronic device, it is still prohibited by this rule. Additionally, it creates a potential distraction to the opposing team.