However, computers scare me to death. I've never seen the inside of a computer.

Nothing to worry about, it's not any worse than a car engine actually. :-D

As for buying a regular hard drive and putting it into a case... I have no clue how to do that.

If you can tell the two ends of a screwdriver apart, you should be able! A HD looks like a big metallic soap bar with 2 connectors on one side, a big one for data, a small one for power. The USB casing (bought empty) is actually an empty box with, inside, a big connector for data and a small one for power. I guess you start to see a pattern... ;-D
Plug the connectors, fix the HD inside the case with 4 screws, close the case (some more screws), and that's it.

The one I have now is an internal hard drive put into an external casing...

Most are. But that's not the problem; As you said, the HDs inside are the same you've got inside your computer, so if they work inside the computer, they should also work inside an external casing (and the other way round).

Back to your problem:

• I think it would be a good idea to ask that nephew of yours to give it a quick look; You might salvage all your files. Tell him to check the HD inside; Maybe it's just your casing which has a problem.

• The USB-1 to USB-2 problem isn't important. USB-2 is just a faster version of USB, and if your computer doesn't support it, it only means your USB drive isn't as fast as it could, that's all.

• To format a drive, you just click with your right mouse buton on it and select "Format...". That DELL CD you use is a complete reinstall of the system, and since your computer is old, it reinstalls an old system, without all the latest fixes, drivers, and most important, security patches! You'll have to make a long session of Windows Update afterwards.