@ChristianRemling: Thank you for your answer.

My suggestion is to use the ideas in the proof for $\ell_\infty/c_0$ to prove the case $\ell_\infty(\Gamma)$, without using the quotient $\ell_\infty(\Gamma)/c_0(\Gamma)$.

In Theorem 2.40 it is proved that every separable subspace of $\ell_\infty/c_0$ is contained in an isometric copy of $\ell_\infty$. Maybe the argument can be adapted to $\ell_\infty(\Gamma)$.

I saw it yesterday. The argument does not prove the isometric case. In fact, if every isometry between separable subspaces of $X$ extends to an isometric isomorphism of $X$ then $X$ is a space of universal disposition, but$\ell_\infty(\Gamma)$ is not (Theorem 3.34). If you need the isometric case, you have to find a different argument.

It would be nice if you write an answer indicating how Szarek's results answer both questions.

@Onur Oktay: I missunderstood your first comment. I thought that only the first question was answered in Szarek's paper.