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So you agree that you need to use another law. The argument can be stated this way: Let us consider a system of particles. According to N1L none of the momentums of each particle remains constant unless forces act on it. Here the forces are all forces of interaction. So the total rate of...

When you use F = ma, N1L is automatically implied, is it not.

I think it would be fair to say that it's a consequence of both first and third laws.

That's an interesting question. I wonder why no one has attempted an answer already. Perhaps the hammer has undergone a 180 degree rotation about an axis perpendicular to both the length and the axis along which it is rotated? In other words, the near end is now the far end?

The point is, you can calculate a maximum angle. For a high enough v it will go over, but you don't know beforehand if it's a high enough v. So you should find the formula for the angle from energy conservation. I suppose that's all the prof wants.

Is it possible to show directly from Newton's Laws that conservation of momentum and energy follow from principles of symmetry?