I have an Xbox One controller, a Dual Shock 4, and various Bluetooth 8bitdo controllers, and all of them seem to work just fine through Proton, as long as the game has gamepad support built in. I don’t have Dark Souls 3 but I’d bet the story would be the same.
No, the original versions of the Mega Drive had no built-in games. They will display nothing but a black screen if a) there is no cartridge inserted or b) the console can’t read the cartridge.