"Does the writer of a song have an exclusive negative right to perform it? That would make all covers copyright violations."

Song writers/authors have what was called (and may still be called) 'first recording rights'. However, after a writer has recorded AND released a song, it can be done by anyone provided the royalties are paid. I don't think that law has changed. It can be done by another provided the author signs away the right to record the song first.