'As well as' can have different meanings. It can be use comparatively, as in "He sings as well as he dances." Or it can be used to mean 'in addition to" as in: "For this part in the play, he must be able to sing as well as dance." Which is a more explicit version of your sentence, "He needs to sing as well as dance." Your first sentence above is incorrect. He sings as well as dancing. XXX Possible corrections depending upon your meaning: 1) He sings as well as he dances. (Means he is as good a dancer as he is a singer. He is equally good in both activities.) 2) He sings as well as dances. (This sentence is a little ambiguous - it could be interpreted as the meaning of the above sentence, or it could be interpreted to mean 'He sings and also dances.' The latter being a less ambiguous way of delivering the intended meaning.

"He can sing as well as dance" would be the correct phrase of the three phrases you listed.