(Edit: I answered Q2 initially, ignoring Q1.)

Q1: The spectral sequence is right except that it starts at $E_1$.

Q2: It is true if $\mathcal{F}$ is unitary in the sense that the underlying representation of $\pi_1(M)$ is unitary, then the spectral sequence degenerates at the $E_1$ page. This seems like a folklore fact, so I would need to think to find a *good* reference. But one that comes to mind is Timmerscheidt's appendix to a paper of Esnault and Viehweg, Inventiones, vol 86, pp 189-194. (He proves something more general, so it might be hard to follow.) By the way, this should be enough to conclude that condition of the last sentence of your Q2 is sufficient.

I don't think degeneration is true for non unitary local systems.
The best result in general is due to Simpson that, at least when $M$ is projective, $H^*(M,\mathcal{F})$ is isomorphic to the hypercohomology of the Higgs complex of the associated Higgs bundle. But perhaps this going too far afield.

**Added** Actually why don't just give you the proof. The key analytic fact that
makes the usual degeneration work is the Kähler identity
$$d d^*+d^*d= 2(\bar \partial \bar\partial^* +\bar\partial^*\bar\partial)$$
This continues to hold for a unitary local system, where $d$ is now the connection. The point is that the arguments needed to establish the usual identity carry over almost word for word when you work with a unitary local frame.