Stopping the lens down in Aperture priority movie mode does increase the dof just as it does for a still photo. However you need to set the aperture prior to turning on Liveview. The aperture won't change to a new setting until you turn off Liveview, reset the aperture and then turn Liveview on again. You can't change the aperture while recording.
The reason the LCD stays bright is the gain to the LCD is increased in order to keep a constant brightness over a wide range of ambient light.
I normally don't shoot videos but of the 3 or 4 I've taken this is the way the camera reacted to different f stops. I only use A mode so I don't know how it would react in P, M or M mode.
I know there are a few on this forum that take videos all the time. They might be able to expand on my experience.