Are Cro-Magnons direct descendants of Neanderthals?

No, they are not direct descendants of Neanderthals. We can say that the Neanderthals were in some way an intermediate link in the development of ancient man. So they appeared about 300 thousand years ago and were already much more advanced than their predecessors Pithecanthropus. Nevertheless, Homo sapiens, who arose about 50-30 thousand years ago, was more intellectually developed in comparison with the Neanderthals. Thus, soon the Cro-Magnons were able to push back the Neanderthals and take dominant positions on earth.