You can identify $$W_qR^qf_*\mathbb{Q}=im[R^qg_*\mathbb{Q}\to R^qf_*\mathbb{Q}]$$ It is enough to check this fibrewise, where it's Deligne's Hodge II, cor 3.2.17. Now compare Leray spectral sequences for $g$ and $f$, and take the image $$im ([E_2(g)\Rightarrow H^*(X)]\to [E_2(f)\Rightarrow H^*(U)])$$ This should give your desired answer to Question 1. [Note: there's a subtle strictness question that I overlooked. I'll try to sort it out when I have more time. ]
For Q2, let's first suppose that $S$ is smooth and proper. Then $H^p(S, W_qR^qf_*\mathbb{Q})$ is pure of weight $p+q$, so $d_2,\ldots$ must be zero because it goes between Hodge structures of different weights. This is just the barest outline, but see my paper for some more details. I think the result is true in general, but you would need to use Saito's version of the decomposition theorem in his category of polarizable Hodge modules. I'll see if I can supply some more precise arguments later on.
Added Note: As Dan noted below, Q2 follows easily from the first paragraph, and Deligne's degeneration argument for $g$.