Rules of Badminton state:
- FAULTS: It shall be a ‘fault’:
...
13.3 if in play, the shuttle:
...
13.3.6 is caught and held on the racket and then slung during the execution of a stroke;
13.3.7 is hit twice in succession by the same player. However, a shuttle hitting the head and the stringed area of the racket in one stroke shall not be a ‘fault’;
13.3.8 is hit by a player and the player’s partner successively;
Source: http://www.bwfbadminton.org/file.aspx?id=680849&dl=1
So to answer your questions:
So, what are the rules regarding this? Does the point count?
The point counts. But it is a fault. Hence, the point is awarded to your opponent. See 13.3.8. However, be award that if the shuttle first hits the head and then the stringed area, it is NOT a fault and therefore a legal shot.
If not, then is the point awarded to the opponent or is the rally abandoned?
See above.
Is there a technical/official term for "double touch" in Badminton?
When looking at the official rules, it seems, there is no official term.