My iPod shuffle won't connect to my windows 10 pc

Since the shuffle works on another computer, it seems to be fine. If you already tried uninstalling iTunes, you probably did not completely uninstall iTunes and related software components

Remove and reinstall iTunes and related software components for Windows 7 and later - Apple Support

Be sure to remove all components of an iTunes installation, in the precise order described. Then, download and install the latest version of iTunes from here

https://www.apple.com/itunes/

Take a look at this article If iTunes doesn't recognize your iPhone, iPad, or iPod - Apple Support

You were using the same USB docking cable (that you use with your PC) when you tried it on your kid's PC?

Then iTunes locks up so I can't eject the iPod...

That may indicate a hardware-related connection issue when you connect it to your PC. The shuffle itself seems to be fine, since it works with the other PC. If possible, get it to appear in iTunes and go to shuffle's Summary settings screen (before the problem occurs). There's a checkbox there for Enable disk use. If that option is not checked, check it and Apply. Is there any difference?

If that's not possible (or it does not affect problem), you can post back and include your shuffle model ("generation"). There are four types. This document shows the shuffle types, if you're not sure

Identify your iPod model - Apple Support

2nd gen shuffle is an older model, from way before USB 3.0 existed. Does your PC have USB 3.0 ports and the other PC (where shuffle works) have USB 2.0 ports? If so, that may be relevant.

Another possible difference is desktop PC versus laptop PC. Is your PC a different type from the other PC? This can affect power management; a laptop may have "energy-saver" features that a desktop PC does not have. If your PC is a laptop, be sure to connect its power adapter (so it's not running on battery power).

Analyze any other differences between your PC and other PC; it may provide a clue for why it works properly with other PC, but not your PC.

Do you happen to own an older USB 2.0 hub that is self-powered? "Self-powered" means it has its own power supply (does not rely on power from USB port). If you have one, as a test, connect USB 2.0 hub to your PC's USB port, and connect shuffle to the hub. Be sure to connect the hub's power supply. Does that make a difference?

You should to do this test with all other USB devices disconnected from your PC. You can leave standard USB keyboard/mouse connected, if used.

It should work with USB 2.0, AND it worked before a few weeks ago... Have you tried connecting shuffle to a USB 3.0 port on the expansion card?

The type of error (the "lock up" of iTunes) seems to indicate something related to hardware, like iTunes losing physical access to the shuffle (in an unusual way) after it initially has access. Normally, losing access to iPod unexpectedly just causes an error message to appear. And you cannot see the shuffle through Windows either, after the problem occurs. Plus unplugging it causes some fatal Windows error.

And, again, the shuffle itself (with the docking cable) is apparently not the cause, because it works with the other PC. Maybe there's a problem on your PC's built-in USB bus. On a Mac, there's a procedure to reset the overall hardware by unplugging it from power. Have you tried shutting down (powering off) the PC, and unplugging it from power for a few minutes?

And unplugging it still causes a fatal error on the PC.

That obviously should not happen. Do you have a regular USB flash drive? If you do, are you able connect it and disconnect it without the fatal error happening?