According the official LAWS OF BADMINTON, set by the Badminton World Federation, chapter 14 is affected: LETS. Here the different situations are outlined when you need to play a let and in paragraph 14.2.3 the following about that is mentioned:
- 14.1 ‘Let’ shall be called by the umpire, or by a player (if there is no umpire), to halt play.
- 14.2 It shall be a ‘let’, if:
- 14.2.1 the server serves before the receiver is ready (Law 9.4);
- 14.2.2 during service, the receiver and the server are both faulted;
- 14.2.3 after the service is returned, the shuttle is:
- 184.108.40.206 caught on the net and remains suspended on its top, or
- 220.127.116.11 after passing over the net is caught on the net;
From your description, it seems as if the second subparagraph is applicable: 18.104.22.168.
As to your second question, it does matter when it happens. The title of 14.2.3 states that it is only valid, when the service is returned. So, if it were to happen during a backhand low service for example, then a point would be given to your opponent.