How do I loosen my bathroom stopper pivot nut
Don't bother with all that, just remove the U shaped pipe (The Trap) which should be a pressure fitting that can be easily unscrewed without any tools. Most of the crap will be in that U part of the pipe, the rest you can get out by sticking something like a toilet brush or pipe cleaner up the exposed pipe.



BTW: Be sure to put a bucket under the pipe before you unscrew it, some nasty water will likely leak out.

As for that pivot nut, if you really want to remove it, it USUALLY is easy enough to unscrew with your bare fingers, but if it is rusted you can use a pair of channel lock pliers. If it is just spinning however, it may be all the way off the threads and you just need to pull it back. It is attached to a rod that pivots in the middle of the pipe. You might need to shake it a little to get it loose, it will also probably help to detach the rod that goes from the sink stopper pull to that nut since it will get in the way.