Two questions--two answers
To the first, If the original singer can demonstrate prior art (in the US, this generally requires a provable written document) the person recording the work has no rights. If this demonstration is not available, the singer's out of luck. The solution, if you're a song writer, is to register the songs yourself.

As to the second question, legally that depends solely upon the contract that defines whatthe purchaser bought. If the seller owned the rights, he can sell them.

After the fact, there's little if anything the original singer can do.