Where does $\dot{\phi}$ come from when deriving $\dot{\hat{\phi}} = - \dot{\phi}\hat{r}$ ?

This is the magnitude of what you get when taking the limit as $\Delta t \rightarrow 0$ of $\frac{\Delta \hat{\phi}}{\Delta t}$. The direction of this limit points radially inward, and magnitude is just the limit as $\Delta t \rightarrow 0$ of $\frac{\Delta \phi}{\Delta t}$, which is $\frac{d \phi}{dt}= \dot{\phi}$. See Fig. 1.13 of your text.