What are these weird symbols in my text?

The symbols look like the "top half integral" character at Unicode (hex) 2320.

I've never heard of Word putting anything like that into a document automatically. Do I understand correctly that this text has never been outside a Word document before the symbols show up?

Exactly. I can open up a new Word document, type in some text, copy and paste it as Unformatted Text, and those weird characters appear. It's really strange.

Yes, it is really strange, and I don't have an explanation.

Here's a troubleshooting technique that may help to eliminate some possibilities. Hold down the Ctrl key while you start Word, and say yes to the prompt, to get it running in safe mode. That causes Word to run without loading Normal.dotm, any add-ins, or any non-default registry entries.

Now try your copy/paste cycle. Do the characters still appear?

If not, then suspicion falls first on add-ins. If there are any templates in the folder %appdata%\Microsoft\Word\Startup, move them to some other folder and restart Word. If the characters haven't returned, you can move the templates back one at a time, restarting Word and retesting after each one.

If it isn't an add-in, then it could be something odd in Normal.dotm. Try renaming that file, restarting Word, and retesting.