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.

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 <a href="http://front.math.ucdavis.edu/0301.5140">paper</a> 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 $g$.