The condition you give is the Holder condition, not the Lipschitz condition as you state in your question. (As Jacob Schlather points out in the comments). It is a more general condition, in that any function which satisfies the Lipschitz condition also satisfies the Holder condition, but not conversely.
However, $1)$ and $2)$ can be proved directly fairly easily (also, $2) \implies 1)$
For part $3)$, can you think of a differentiable function which grows faster than any fixed polynomial? (this property means that it cannot be in $F$)
Edit: As Jonas Meyer says in the comments, any polynomial with degree greater than $2$ will provide a counterexample to part $3)$. In fact, from part $2)$ we can make the much more general statement that any differentiable function which is not uniformly continuous will provide a counterexample.
For part $4)$, think about $|x|$ for example.