From my limited scientific knowledge
Most lights flash. We don't normally see it though as they're flahing very quickly and our eyes filter it out so we see a solid light. A camera, which catches single frames of time, can sometimes catch it when the light is "off", or refreshing, so it looks like it's flashing when viewed as a video. It's a similar thing to refresh rates on a monitor.
Probably...
Yes, this happens with LED lights. The older bulb lights are just being switched on or off and thus don't flash.
LED lights however are pulsed for a different brightness, so when filmed with a camera which captures discrete frames, these lights can flash because the camera catches them sometimes on and sometimes off, as daan explained. Example: you have one set of lights for driving and braking. When you do not brake, the LEDs are pulsed. This happens so quickly that the human eye can not distinguish the on-off-periods, so you just see a reduced brightness. When you brake and the lights come on, the LEDs are simply being tuned on all the time, so the light appears brighter to the eye.
They're not flickering. It's the recording device.
The answer is above....why are they flickering during recording though?
Sure. All they have to do is make cameras that record at either the same number of frames a second as the frequency of the electricity used. Which only means redefining world video standards or world electronics standards...Is it possible for the camcorder companies to fix this issue?!