Yes (which is for all intents and purposes a no) 🙂
Yes to mechanically propelled vehicles (MPV’s) on motorised routes – for all the reasons tinas and timmys covered.
Strictly speaking bikes have to stick to the Highway code on Bridleways as they are still highways (and a road #), but because there are so few restrictions (the only ones that come to mind are that you *must* obay traffic signs / traffic signals / lights – which you don’t get on Bridleways), it’s effectively no.
Again, the only thing that comes to mind, is you must not “ride on a road in a dangerous, careless or inconsiderate manner”, with ‘road’ including Bridleways (#). A quick google found this, so a fine at worst (two case examples)
Plus as has been mentioned earlier the HC isn’t the law as such, but an easily digestible summary of the various laws in a single booklet.
It starts to get complicated in scenarious tinas touches on (what is a “road”?, if not a “road” is it a public place, etc etc – thankfully it’s OT though 🙂 )