Is there any description of unital idempotent ($F^2(x)=F(x)$) morphisms of a von Neumann algebra into itself? Or, equivalently, of weakly closed subalgebras which are retracts as von Neumann algebras?

Yes. The kernel of F is an ultraweakly closed *ideal of M generated by some central projection z. M splits as a direct sum of zM and (1z)M. As a 2x2 matrix F has only two nonzero entries, one that corresponds to an idempotent automorphism (hence the identity map) of (1z)M and another one to an arbitrary morphism from (1z)M to zM. Thus idempotent morphisms are classified by central projections and morphisms from (1z)M to zM. 

