Mine stay on while I'm driving but are very dim, its nice though because its kind of like ambient lighting. They do get brighter when you open the door too of course. It also may be because my LED's are 1w but I'm not sure.
kinda, in the middle position where its fully on when you open the doors and such, it will always have a slight glow, but if its in the off position there is none, and no matter what position when you lock your car and arm the alarm they fully shut off