Crossing the net with the racket after hitting it on your side is allowed, as laid out in §13.4.2 of the laws of badminton:
It shall be a fault, if in play, a player invades an opponent's court over the net with racket or person except that the striker may follow the shuttle over the net with the racket in the course of a stroke after the initial point of
No, an idle racket will not lose tension as quickly. Any impact sustained by a string-bed will reduce its tension. A more intense impact will reduce the tension more. Thus, the effect in badminton is weaker than in tennis, but still existent.