Install Receptacle in damaged drywall
Don't "attach to drywall" if a stud is available.

Seriously.  You can have some pretty large insert/remove forces on a socket - 10 or 20 pounds.  Drywall is chalk covered with paper. It is simply not made to bear that, and the goofy "wings" that grab-the-drywall boxes use won't hold up to the strain.   

If it was a switch, maybe not so bad... but for a recep, no, definitely use any means necessary to screw into that stud with a box capable of that. 

I would go for a steel box because it will attach well to screws with no risk of tearing out... and then use wood screws with either a 90 degree ratchet screwdriver, or a hex head screw and a wrench.  Royal PITA, once, and then the job is done first-rate and you never have to sweat it again.  

Now that we're not using it for attachment, that drywall damage is just an "oops". Cover that with an  extended cover plate, which are widely sold for just that purpose.