Can つよ be used to describe a person's physical or mental strength as well?
As far as I know, yes. つよい (強い) indicates something is strong, powerful, etc. You can definitely describe someone as つよい to mean they are physically strong. I do not know about using it for mental strength, but I guess it would also be acceptable.
whats the difference between tsuyoku and tsuyoi?
Tsuyoku is an adverb, tsuyoi an adjective. I hope that helps.
I will always remember つよい from watching Yowamushi Pedal. The captain of the main opposing team said it A LOT.