Is a Lipschitz function differentiable?

It is not always true indeed, good counterexample could be $x\mapsto |x-a|$. But rather, we have

Theorem: Radamacher theorem says every Lipschitz function is almost everywhere differentiable

Fine a nice proof of this theorem here: An Elementary Proof of Rademacher's Theorem - James Murphy or Here using distribution theory

The function $$x \mapsto \left|x\right|$$ is Lipschitz-continuous (with $k=1$) but not differentiable at $0$.

No, it does not imply $f$ is differentiable.

Try $f(x) = |x|$ as an example!