More directly if the game engine only updates player state every 60 seconds (tick rate) then is this 4ms advantage actually present for the 240Hz case?
Further if your network has more than 4ms of jitter then I don't think you can make any concrete claim in either direction.
My theory about why it helps gamers to have a higher frame rate is that for something like a whip turn, with a low frame rate, your brain has to take a brief moment to work out where it ended up looking after the pan. But if your frame rate is high enough, you brain can keep updating its state during the pan because the updates are continuous enough not to lose “state” during it. This means when you finish the fast move, there is no delay while you reorient yourself for a few milliseconds.
Yes, because the visual feedback on your client matters as well (faster reaction times).
Also, it means each frame is more recent. Also, higher refesh rates reduce the mean variance between the timestamps of rendered frames offered by the GPU that are drawn on the monitor, meaning the visuals are smoother. The smoothness is pleasing to the eye and makes it easier to focus.
Further if your network has more than 4ms of jitter then I don't think you can make any concrete claim in either direction.