It's a logitec Dual Rumble.
It has worked on every game that supports USB controllers that I've played.
But I'm saying that the ability to select 'gamepad' from the controls menu, won't work.
It's the first option. You go to it, you try to select it by clicking on it and it just makes a noise and nothing.
It won't toggle to 'on'.
I use windows 7.
Unfortunately, I'm not tech savvy enough to help you with this, but hopefully the info you provided will be enough for someone else more in the know to help you sort out your issue.
The build of Nuclear Throne GOG has right now doesn't support daily missions and challenges, but those will come eventually, so, if it is, in fact, a problem with the game, I'm pretty sure it'll get fixed any time soon.
I don't think it's an issue with your OS, since Windows 7 is a supported operative system, everything should be fine on that end. As for the controller, I really don't know. I can safely say the GOG build works with a PS3 controller with no issues whatsoever, so maybe it's a controller thing; perhaps toggling the gamepad on via menu isn't working for you because the game doesn't recognize your Logitech as a valid controller. Like I said, try using the hotkey instead of the menu; according to the official wiki for Nuclear Throne, maintained by Vlambeer themselves, G should be the key to toggle gamepad on and off without having to go into the options menu. Maybe that works for you?
I would wait for a reply by someone other than me, hopefully they'll be able to address this specific problem and try to solve it for you. Another option is to submit a trouble ticket to GOG Support, they're definitely more in the know than us forum users.