How Do I Loosen A Stubborn, Old, Rusty Wing Nut Bolt Connecting The Toilet Tank To The Seat?

Possibly:
Soak the threads with WD 40 and let sit for 30 minutes or so.
 
then, if there is enough bolt sticking out below the nut, grasp it with vice grips and loosen the nut a few threads or as much as you can get.  if you loosen somewhat, you can probably get vice grip hold on top of bolt, and loosen the nut some more.  Then, pull down on the bolt with grips and you should have enough room to get a hacksaw or bolt cutter to sever the bolt above the nut.
 
Also, you may be able to break the bolt once you get it loosened, especially if it is badly rusted, by holding the bolt securely from the top, and bending the bolt back and forth being careful of the porcelain.

If you have exhausted all other methods you can use a nut splitter. You will need to get new nuts but that might not be a bad idea anyways.

The bolts on your toilet seat are most likely reverse threads. Which means you have to unscrew them by turning clockwise instead of counterclockwise.
:0) Been there, done that.

You will better off if you ring the nut off, if the bolts are that rusty they need replacing.

You can try spraying WD40 on the nut and bolt, let it sit for a few minutes before trying to loosen it again.

You might try some xanax!  Just kidding.  I know how frustrating this problem can be. Try calling a Hardware store.  They can probably advise you what to use.  Good luck.

Go to the auto part store and pick up some nuts,bults spray remover and that will take care of it. Something that is not petroleum base.