It works on my Razer mouse that has a freaking 12 number keypad on the side, so I'd bet heavily, and there's a free demo.

Yes. It works on all buttons and scrollwheels on all mice I have ever tried (including several Logitech ones).

Some mice will have buttons that toggle lighting or tracking speed, I have a feeling these are the kinds of buttons that might not work with the app. Because they are just local buttons, they don't get sent to the OS.

It’s varied a lot by version of the software, and by iterations of the mouse models with such functionality. I recall at least one Logitech mouse where the buttons couldn’t ever be reprogrammed; at least one other where it was unsupported before a SteerMouse update; and others where they were supported like any other mouse button on arrival.

Edit to add that IME most have fallen into the last category for a several years now.

It does on my MX Vertical. It also lets you set different actions per app. I've used that to bind the forward and backward buttons to do the right thing in most apps I touch so I have a universal shortcut for that navigation.