It's designed around NES-style controls: directional keys, confirm, cancel, menu. Any controller should do once you bind the keys (not configurable in-game, as far as I can tell). I've been playing it with a Wiimote, myself.
I played 99% of the game with a wireless 360 gamepad.
There's a name entry field that you'll want a keyboard for, but other than that I can't think of anything that'd get in the way.
There's a couple places where you might need to get off the couch and interact with the computer itself briefly, but I don't know how to say anything more about those without spoiling things.
Button config only appears at the title screen, and only if a controller is already detected. I played with the 360 wireless + dongle, but your wired setup should work AFAIK.
Yes, but it's pretty wonky. IIRC, you have to turn the sensitivity all the way down or else joystick movement tends to not reset itself when you lay off the stick.
My Xbox One controller didn't work until I unplugged my joystick (for Elite Dangerous). As soon as I did that the game recognised my gamepad. So try unplugging some stuff if need be.