Is it compatible and customization with the game via menu settings? And does the it work without having to use 3rd party applications?

Both KotOR games on GOG do not natively support controllers at all. The PC releases of the games have been heavily optimized for keyboard/mouse controls and lack all of the control options of the console versions. Even using a third party keymapper is difficult because of how things like combat management are made to work with m/k. The Steam version of KotOR 2 has been updated with added gamepad support by Aspyr (developer of the Mac and mobile ports of the game), but that update has not come to GOG yet and they have not updated the first game at all.