For the second one, note in the Fourier expansion of $\mathbf{A}(t,\mathbf{r}) = \sum_{\mathbf{k}} \mathbf{A}_{\mathbf{k}} e^{i (\mathbf{k} \cdot \mathbf{r} - \omega t)}$ that $\mathbf{k}$ determines the direction the wave moves in, it acts like a velocity in the direction of motion as you can see from the units alone or an argument like
this.