My first question is the following:

Q1: Let $X$ be a Banach space. If its dual $X^*$ is weak* separable, does $X$ admit an infinite-dimensional and separable queotient $X/M$?

To the best of my knowledge, the dual $X^{*}$ is weak* separable, when $X$ satisfies one of the following:

(i) $X$ is separable;

(ii) $X$ is the dual of a separable Banach space;

(iii) $X$ is Hereditary Indecomposable Banach space. That is, every infinite-dimensional closed subspace of $X$ can not be written as a direct sum of two infinite-dimensional closed subspaces.

And, I see that if $X$ satisfies (i) or (ii), $X$ admit an infinite-dimensional and separable queotient.

Q2: Is it true that $X$ admit an infinite-dimensional and separable queotient, if $X$ satisfies (iii) ?