sincerely curious — does Linux natively map the numeric keypad on my mouse? Is there a GUI?

It depends from the quality of your mouse. There are good quality keypad mice, where the numeric keypad presents itself in hardware as a proper USB keyboard endpoint. Those just work, with more than just Linux moreover.

Then there are the poor quality keypad mice where the keys may be engraved as numeric keys, but on the wire the device does things like not present them as keys on a keyboard at all (sometimes as lots of high-numbered mouse buttons instead, sometimes as something else), or present them as keys but using reserved manufacturer-private values instead of the standard codes for [1/Ins], [2/Down], and so forth. These won't work out of the box with anything.

Most office-use "numpad mouse" devices are the former in my experience (although they sometimes do bad things when it comes to NumLock). Alas, there are "gaming mouse" devices that are the latter.

It does not. I guess my devices aren't exotic enough.

https://wiki.archlinux.org/title/Razer_peripherals

I think things like extra buttons work fine. But if it gets too far out there, you have to get something that someone reverse engineered. Because proprietary software and devices are the norm, unfortunately.