About the rule "Only one action can be mapped to any given button. For example you cannot bind Up and Right to the same button."

I use a Logitech F510 as my controller, and it seems like it registers Up-Right as a separate button -- pressing the D-pad that way hits neither the Up button nor the Right button. In order to make my controller act like an SNES controller I have to bind those functions like this:

Am I allowed to use this controller, or would I need to go another route, like playing with my keyboard?

That is fine, and is a known limitation of modern logitech pads. It's intended, I believe, to improve control for fighting games but sucks for real retro games. Just be aware that this will possibly make movement very difficult at a higher level, and might make pumping near impossible (if you ever bother with it.). The main idea behind the rule is that, when playing on keyboard, you can't bind diagonals to make clipping super easy.