  We address some open problems concerning Banach spaces of real-valued
Lipschitz functions. Specifically, we prove that the diameter two properties
differ from their weak-star counterparts in these spaces. In particular, we
establish the existence of dual Banach spaces lacking the symmetric strong
diameter two property but possessing its weak-star counterpart. We show that
there exists an octahedral Lipschitz-free space whose bidual is not octahedral.
Furthermore, we prove that the Banach space of real-valued Lipschitz functions
from any infinite subset of $\ell_1$ possesses the symmetric strong diameter
two property. These results are achieved by introducing new sufficient
conditions, providing new examples and clarifying the status of known ones.
