Yea, it's baffling that the abstract doesn't mention post-quantum crypto and make some sort of argument that it's somehow not vastly easier to implement.

SIDH and New Hope are not signature schemes.

How do they work? Are they just difficult to write quantum algorithms for or is there some fundamental mathematics involved that stops a quantum computer enumerating the solutions?

To the best of my knowledge there's not even any known mathematical theorem that would prevent such algorithms on regular computers. I don't see how this would be any different for quantum computers.

That is correct. Any proof that digital signatures are secure on classical computers would imply P != NP.

Further, it is known that BQP (bounded error quantum polynomial time. The quantum computer equivalent of P) is contained in PSPACE. However P=PSPACE is still an open question. This means that we still do not have any theoretical proof that quantum computers are more powerful that classical computers.