That claiming a false mileage when advertising a car, or on an MOT certificate is not a DVSA issue, it's trading standards. A very suspect low mileage and iffy progresssion on a car I saw online.
Secondly, that if you sell a car privately and return the V5, then DVSA automatically refunds the tax, had a cheque in the post this morning. Previously It'd always been a case of take the tax disc back to the post office in my experience