In fairness to people who ride quad bikes, and no I'm not one of them, they do not look like quad tracks at all. Quads, the types ridden illegaly by chavs, generally have a solid rear axle which means that they cannot dig deep ruts without all progress being halted. Agricultural type quads tend to be much more expensive, and whilst they can dig ruts quite well, there are not so many about.
I would think that it's more likely that that mess has been created by a four wheel drive or farm vehicle, could well be that it's the land owner just going about his business.
It seems that the problem in that first picture is inadequate drainage which allows that kind of surface damage to be created easily.
Have you called your local ROW officer and let him/her know ?
As they de-classified old roads, i don't think it will have a land owner, they are marked up as green bridleways, which are right of way, as opposed to brown bridleways which would be private land with permission.
You could be right about it being 4x4 tracks, i don't know enough about it, but i can say they are very very deep in places.
In any case, they are no longer roads, but bridleways, so there shouldn't be any motorised vehicles on em. It is so bad in places that you need a horse or 4x4 to get through it
edit. ah "Right if way", where would I get the number from?