Removing Lug Nut from Spinning Stud

If you're going to remove the hub/wheel from the axle, leave the wheel on the ground and jack up the other rear wheel until the axle/hub separate.
 
This is getting out of control with great ideas.
Drilling it out so the wheel can be removed is a simple and inexpensive job. If he doesn't have drill bits, he's not going to have all the other niceties mentioned. Buy some decent bits and don't run them so fast. Learn how to use them properly. There are important details to that. And use cutting oil... any oil.
This type of tool purchase will continue to be useful for the rest of your life on many other jobs so it's not a one job deal.

If it's spinning on you, you will likely have to remove the wheel, drum, and hub as an assembly as mentioned and work from the back. You might be able to remove the hub with an axle nut or two protecting the threads. Flip one over(castellations in) and leave it slightly above the threads. Pop on it with a heavy hammer. Sometimes pulling on the tire, wheel or even lightly prying on the hub will help offset the blows. Be gentle with the prying though.

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Made some progress yesterday. One thought was to drill the side of the nut and try to twist it or break it off. Then I realized that the shoulder at the end of the nut would be impossible to get to with it on the wheel. So I started to drill the center.

I used a nice set of Bosch drill bits and plenty of quality cutting oil every 15 seconds of drilling or so. Both of these made a great difference in cutting progress.
 

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Victory at last. I attached some images of the results.
 

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I also noticed that when I took the hub off that it had this weird ridge in it. Is the hub bad or is this okay to reuse?
 

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So it looks like you were able to save the wheel. I'd be more worried that the hub was still flat than having a gouge especially after all of the work to get the wheel off. How deep is that mark? If the stud was spinning you won't likely be able to reuse the hub anyway. Unless you drilled it and resized the hole to a larger stud, but then you'd have one different size lug and lug nut. I'd say replace.
 
If the new stud still has a good fit in the hub hole I would not worry about the scar on the back side of the hub as it does not look very deep
 
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