Do maps have flows?

In general, there are obstructions for a map being the "time one map" of a flow (see this question).

However, and I am not quite sure this is what you are looking for, there is a general procedure to construct flows out of maps, namely the suspention. For the map $g:M \to M$ you consider the flow in $M\times \mathbb{R}$ given by $\varphi_t((x,s))= (x, s+t)$ and you quotient by $(x,s) \sim (g(x),s+1)$. This gives a flow, whose time one map preserves a set homeomophic to $M$ where the dynamics is $g$.

In the case of a map from $[0,\infty)$ to $[0,\infty)$ fixing $0$ you can work out flow to be defined in $\mathbb{C}$ and such that the time one map is $g$, so this would give a partial answer to $2.$.