Why not у кухні? Is this just an exception?
same as на балконі. I guess it is an exception. At least I do not recall any rules related to this.
I think кухні also means food/cuisine?
In nominative it's кухня and yes, it does also mean "cuisine"
The short for Холодильник doesn't exist?
If it does, it is not widespread,
Absolutely. Thank you for your reply!