Is every element of $Mod(S_{g,1})$ a composition of right handed Dehn twists?

No. A. Wand: Factorisation of Surface Diffeomorphisms discusses conditions when a mapping class is a product of right-handed Dehn twists. Theorem 5.2 of that paper gives an explicit counterexample to your question.

Yes. As pointed out by Ian Agol, I actually answered my question.

I observed that the identity is a non-empty composition of right handed Dehn twists in $Mod(S_{g,1})$. A priori this is not trivial. I was thinking about monodromies on Brieskorn-Pham singularities $(x^p+y^q)$ which are freely periodic and a composition of right-handed Dehn twists (by morsifying the singularity). One can easily see that this solves the problem in the cases that I was asking originally since all surfaces $S_{g,1}$ appear as Milnor fibers of such singularities for all $g$.

EDIT: I removed the last part of the answer since it was not true in that generality and this answers completely what I wanted.