If $d$ is a separable metric on a Banach space, approximate a $d$-Lipschitz function by Fréchet differentiable $d$-Lipschitz functions

Question

Let $(E,d)$ be a complete separable metric space, $$|f|_{\operatorname{Lip}(d)}:=\sup_{\substack{x,\:y\:\in\:E\\x\:\ne\:y}}\frac{|f(x)-f(y)|}{d(x,y)}\;\;\;\text{for }f:E\to\mathbb R,$$ $\mu$ be a signed measure on $(E,\mathcal B(E))$ with $\mu(E)=0$ and $$\int d(\;\cdot\;,x_0)\:{\rm d}|\mu|<\infty\tag1$$ for some (hence all) $x_0\in E$ and $$\operatorname W:=\sup_{\substack{f\::\:E\:\to\:\mathbb R\\|f|_{\operatorname{Lip}(d)}\:\le\:1}}\int f\:{\rm d}\mu.\tag2$$

How can we show the following claims?

1. We can equivalently take the supremum in $(2)$ over those $f$ which are additionally bounded.
2. If $E$ is a $\mathbb R$-Banach space (and $d$ is any separable metric on $E$), we can equivalently take the supremum in $(2)$ over those $f$ which are additionally Fréchet differentiable.

Regarding 2: The crucial ingredient should be the separability of $d$. I guess some kind of mollification could work.

BTW: Can we replace the integrals in the definition of $\operatorname W$ by their absolute value?