Which complete orthomodular lattices arise from von Neumann algebras?

Question 1: Yes, if you take the von Neumann algebra morphisms to be normal $*$-homomorphisms. Restricting any such map to the projections will preserve sups and orthocomplements.

Question 2: No, this already fails in the commutative case. Look up "abelian AW*-algebra" or see the discussion of Stonean spaces in volume one of Kadison-Ringrose.

Question 3: (a) Any von Neumann algebra is generated by its projections, so the functor is faithful. It is not full, just look at $M_2(\mathbb{C})$ where the lattice of projections contains a $0$ and a $1$ and $2^{\aleph_0}$ incomparable elements between those two. There are all kinds of morphisms from this lattice to itself that don't extend linearly.

Question 3: (b) I'm weak on category theory, but I'd imagine the $M_2(\mathbb{C})$ example falsifies both possibilities.

Question 3: (c) I don't think there's any good answer to this question, and that's the reason research in this direction petered out. The idea of asking for a full set of normal states is good, and this does answer the question in the abelian case (again, see Kadison-Ringrose), but I doubt it will work in general. There's just no way to linearize these states on lattices. But counterexamples will take some work.