yes all cat C once vic checked has a marker on the V5, saying that this vehicle has been subject to an insurance claim which may of resulted in major damage, or words to that effect
but beware, many large lease/hire companies and/or corperate companies only have third party insurance, so any accident damaged vehicles are sold off to salvage dealers at a higher price because they wont show up on the HPI check, just take a look on ebay at repairable salvage, many say non recorded
I don't think this is possible
If you are brave enough, a Cat C can be the bargain of the century. Now-a-days, just having the airbags go off in a car can make it a Cat C as it now required a new dash on most modern cars. I have a friend with a 2013 - 63 reg Nissan Juke which he only paid a total of £3500 for the car and repairs he carried out in his shed to the dash and small damage to front of vehicle. He saved around £10000 in buying one as a complete used car!