I know that eDuke32 does.

That statement is true for newer, XInput PC games. For older DirectInput era PC games (like most of GOGs older Windows games), the XBox360 controller works less great requiring at least some fiddling, and a DirectInput gamepad is the preferred option that usually works out of the box. Logitech F310/F510/F710 is exceptional gamepad in that it offers both modes. For instance regarding the four Prince of Persia games that GOG sells, I think with the older three games I like to use F310 in the legacy DirectInput mode, and the newest PoP game in XInput (X360 gamepad) mode. Then it works pretty much without any fiddling and adjustment with these games.

Also note that many of those old PC games (specifically in the first-person shooter genre) were optimized for mouse+keyboard gameplay. Some of them might in practice require the extra precision and speed that e.g. mouseaim grants, with a gamepad you might find the gameplay a bit too hectic with them. GOG doesn't add specific XInput gamepad support for those games, only makes them more compatible with newer Windows versions and graphics cards.

But it depends. I originally played e.g. MDK on PC with keyboard and mouse, many many years ago. It was fine. But later I played the GOG version and found out I was able to remap the controls satisfactorily to the gamepad (F310 in DirectInput mode, I think), and the game works great with the gamepad, mainly because it doesn't need precision aiming (it has heavy autoaim, except when sniping), and with a gamepad I was also able to get analog controls for movement, something I didn't get with a keyboard, naturally.

I'm using "xbox360ce" with my usb gamepad. It's working like charm. You can use this app with your driving wheel too, also it's open source app.

Dosbox have mapper with ctrl+f1 combination