Hi lajevardi,
I'm sorry to hear that your wife is having such terrible pain. You said your dentist did several tests, and if there was no reaction to those I would guess (and it is only a guess; I am by no means a professional) that it's not a tooth causing the problem.
You mentioned that she had her sidom teeth removed awhile ago. When I had mine removed, earlier this year, one of the concerns that I was told about had to do with the nerve in the jaw bone. I was told that removing the teeth on the bottom can bump the nerve, possibly causing pain and/or numbness.
Is there a possibility that one of the nerves on that side was damaged, and that's what's causing the pain? This would make sense with her not being able to pinpoint a tooth location and the pain radiating back into the head. I don't know how mouthwashes would help with this (unless the nerve is exposed somewhere and the mouthwash has some kind of topical numbing agent that I'm not aware of, so it numbs for a few seconds, relieving the pain).
Obviously, this is all just a guess, but it might be something to look into.
The only other thing I could suggest would be the possibility of a tooth 'dying' indicating the need for a root canal, but you said that she'd already had the hot/cold sensitivity test, which is the main one for testing for that.
Good luck; keep us posted.