I've heard, mate-of-a-mate style, of a cyclist being given 3 points for doing over 30mph. I've also read 3rd hand accounts of people being banned from driving for "cycling under the influence".
Haven't rode for about 8 years. However, being older and (slightly) wiser, if I was to give the bike a service and take it out on the road I would fully expect to a fine and receive points on my driving licence if found to have riden in a manor falling foul of the law, e.g. ignoring red lights, cycling without bell/lights at night.
Currently, vehicles that don't legally require a licence plate (i.e. bicycles, mobility scooters etc) do not need to pay "road tax" nor have insurance. (I know the previous Labour government said they were going to look into legislating mobility scooters after one rider killed a pedestrian.) As previously mentioned they would fall into the zero VED rate under the current system anyway. However, from what I'm aware of, it's council tax that pays for the bulk of the local road maintainance these days. VED instead goes into the big pot of "general taxation" that, in part, pays for the highways agency, DVLA etc.
Obviously feel free to correct me on any of this...