Correct expression for what?
The average velocity is usually defined as ratio of displacement and the time interval the dispalcement happened. ##\vec{V}_{av}=\frac{\Delta\vec{r}}{\Delta t}##. If you want to express the displacement in polar coordinates you have to define what do you mean by...
The voltage you apply cannot accelerate the electrons in the valence band, so it does not transfer energy to them. This why they are not conduction electrons.
Strong electric field may alter the band structure itself though and maybe reduce the band gap.
This is not the change in magnitude but the magnitude of the change in the vector. Constant magnitude means that the change can be only a rotation . ##\omega t ## is the rotation angle.
This is what I understood by frequency in my post. Not the "the rate of emission of photons" which is related to intensity. I have never seen the 'frequency of light" to refer to rate of emission of photons.
If you have the same intensity and higher frequency there are fewer photons reaching the surface in any time interval. So, fewer electrons are ejected. Saturation current means that all the ejected electrons are collected. Their KE does not matter. But their number does.
The drawing is just a schetch of the principle. The actual charge desnity will depend on the actual sizes and shapes of the components. You cannot get it from general principles. It's the same idea as knowing the actual map of the potential in the system. If you know how thw potential changes...