English question - please help?

Fully it's: "I can neither sing nor dance." Most people would say "I can't or dance."

"I can't sing and dance" would only make sense if followed by "at the same time."

The same rules apply to the sporting example.

"I can't sing nor dance" sounds extremely awkward to me. I really can't imagine anyone speaking that way.

Well.. it depends on whether you are talking about having the skills to play, or being available to play.

You should say "I can neither play basketball nor football" if you mean you don't have the skills to play either

"I can't play basketball and football" means that you can't play both at the same time (sounds funny)

"I can't play basketball or football" could mean that you don't have the skills to play NOR the time to play.

Hope this helps!

It's
I can't sing nor dance

That's proper way but you could get away with saying 'or'
:)