Are amines amphoteric?
Amines do not react with hydroxide to give amide ions - not significantly at least.

The typical $\mathrm pK_\mathrm a$ of an amine is $\sim 30$ (see Evans pKa table) and that of $\ce{H2O}$ is $\sim 15$. The equilibrium constant for the second equation you wrote is therefore on the order of $10^{-15}$.

In order to generate any amide ions you need a much stronger base than simply hydroxide. A typical choice is an alkyllithium (LDA is commonly made by reacting diisopropylamine with BuLi).

Back to the question whether amines are amphoteric: If your definition of an amphoteric compound is simply "can react either as an acid or a base", then yes, it technically is, since it reacts both with acids ($\ce{HCl}$) or bases ($\ce{RLi}$).

However, it may be useful to draw the line at reacting with aqueous acid and aqueous base. Otherwise, nearly everything with a hydrogen can be considered to be amphoteric (for example, $\ce{HF}$ can technically act as a base to form $\ce{H2F+}$). Under this definition amines are not amphoteric.