Generally-speaking, USB mice/trackballs don't need drivers. USB input devices look pretty much the same to the OS. You'd want some kind of non-device-specific program to perform macros when you hit particular buttons.
I have not used it myself, but I understand that antimicrox is one such program. I use Debian Linux, and it's packaged there.
kagis
On the G13, the Arch wiki has some discussion. It sounds like the best option there is "g13d". g13d is not packaged in Debian Linux, and would need to be manually compiled and installed if your Linux distro doesn't package it.
I have no idea what the state of software is to display anything useful on the thing. The Arch page says that the g13d daemon displays a logo on it when it starts up, so it clearly can display things, but I don't see any other functionality it provides (e.g. a clock) or what other software there is that can talk to g13d. If you have any Windows games that have integration with it and can display something on it, if that's a thing, they probably won't know about g13d.
https://github.com/brittyazel/g13d
From a ten-second skim, I also don't know whether the thing is set up to modify functionality based on the particular program with a foreground window, which I don't know if is important, if you want game-specific bindings.
EDIT: The G13 also includes a 160×43 pixel monochrome LCD.
There are a couple of software packages I know of that are intended to display various sorts of system information on small, external LCD displays; these would typically be set up on the front of a desktop computer case. It might be possible, if you have the technical chops, to rig one of these up to the G13's display, as they're already intended to display a small amount of information on a low-resolution display.
This includes lcdproc, lcd4linux, and a few others. They won't be oriented towards extracting and displaying data from a running video game, though, if the G13 does that.
EDIT2: Michael Larabel, who runs the Phoronix Linux gaming site, reviewed using the G13 under Linux back in 2009, and at least at that time, it didn't have native support from lcdproc (and he also commented on how that might be useful).
https://www.phoronix.com/review/logitech_g13/3
As far as how the device worked once we got it functioning under Ubuntu Linux, well, the LCD screen is nice (though with LCDproc support it would be a lot better and more useful) and using this 22-key game pad was nicer for gaming than on a laptop keyboard.