I wasn't very clear about the factorizations (and I am still not in the edited version but I did include a reference) - but as an example of what I mean the mod 3 Moore spectrum S/3 is killed by 3 so we get for X-> S/3 a factorization via X/3. But even though it is 3 torsion it is not true necessarily that the cone on X/3 -> S/3 is still annihilated by 3 - one needs to take a higher power. This sort of thing can't happen in algebraic categories and I wondered (although sort of doubt) if it could be related.

I'll edit in some more details later tonight when I get a chance.

A reference is Hartshorne Ch III Prop 10.4 (the proof of (iii) implies (i)). The basic idea is that identifying the stalks of the cotangent sheaf with m/m^2 one can lift a basis to a regular sequence at x and f(x) and injectivity says the sequence at f(x) maps injectively into the one at x. One can take the quotient at f(x) by the max ideal and by the corresponding image at x which is flat since the quotient at f(x) is the residue field. Then one uses the local criterion to finish by induction.

Good point, didn't think that one through.

You are right... I fixed the answer up - in particular no need to mention affines here and the answer is yes.