How is the direction of drag in the $y$ direction decided? In the lecture the $y$-components of $\vec{f}$ and $m\vec{g}$ should be the same (both down) but we write them opposite in $m\ddot{y} = mg-bv_y$. Why?

So, gravity is always down (positive $y$ in the downward $y$-axis coordinate system we chose). But the $y$-component of the drag force can be positive or negative, depending on whether the $v_y$ is negative or positive, respectively (the diagram in class was just an example of positive $v_y$; $v_y$ could be down, too). The term $-bv_y$ on the right hand side of the EOM takes this into account automatically: if $v_y$ is positive, $-bv_y$ is negative, and if $v_y$ is negative, $-bv_y$ is positive.