The AC compressor is underneath the therm housing, but I don't remember it being in the way. It may be though.
To get at it with vise grips you'll have to remove the AC compressor. Don't forget the lines have freon in them, so be careful with them.
I'm not sure what the silver round mechanical thing is, but if it's in the way, you probably need to move it.....
I would start with a 10MM deep socket. Slide it on and make SURE it's on the flats of the nut, then whack the end of the socket with a hammer to seat it on the socket.
Then, taking your time to make SURE you're pulling straight and not crooked, try to loosen it.
Don't be surprised if the stud comes off as well. No big deal if it does, you'll just need to replace the stud and nut.
Do it right, if you screw up the stud hole you're going to need to replace your water pump [that's what the stud threads into]. That means doing your timing belt, and it doesn't sound like you're ready for that yet.