Is a stolen and recovered bike always a stolen and recovered bike..?

If a claim hasn't been made then it will not appear as a stolen/recovered bike on anything except the police files, and Joe Public doesn't have access to those. It's always worthwhile informing the Police that the bike has been recovered, or you might get a bit of a surprise when you take it for a spin and get tugged.

It won't show up as an insurance loss if it has not had a claim made against it so the only way for it to be on a HPi register is if it still has out standing finance on it.
cat C means repairable but there might be some structural problems, ie bent forks, twisted frame etc.
cat D is just cosmetic, ie broken headlight, scratched fairings etc.
both easily fixed by the likes of us using second hand parts from ebay and the like but the insurance pay dealership prices so factor in the dealership labour rates and not worth it for then to repair.