If they wrote the car off and paid you for it, the entire car is theirs including the tax disk, removing it would be theft. It is also completely down to the insurance company whether or not you get to keep the salvage, and how much they will take out of your payout for it.
If you believe the compensation was inadequate, it's up to you to show the insurance company that you would be unable to purchase a car in similar condition. If you can provide solid evidence of this the offer will usually be raised.
As for NCB protection, if you ended up with the same number of No Claims years, then it has done it's job. Did you honestly believe it was a "No Premium Increase Protection"?
Trouble is, they took the car away before I had even agreed to scrap it or whatever - I went to pick the car up after discussing with the assessor that I just wanted him to look and tell me if there was anything more than superficial damage (there wasn't) and when I got there it had already been removed to another garage prior to going to the storage depot. At the time the vehicle still belonged to me and no monies had been agreed upon or received by myself.
When I was told I wouldn't get the vehicle back I enquired as to the value of replacing the car I was given the price of a grotty rot box from a nearby garage and told it was fair value for my vehicles age. Despite numerous other vehicles shown to compare the unfairness of the value I was offered, they refused to budge.
Then to top it off, the insurance company tried to tell me that if I took the money offered, I would then loose my supposedly protected ncb and my premium would return to what it had been when I first learned to drive at the age of 17 - a hike of about 400 quid per year. It was either that or they generously offered to let me keep my ncb at the expense of halving the money they were to pay out for the car being written off. I was also informed that at no point would I be able to buy back the car and repair the damage to the front wing and bumper/headlight assembly, as it was not allowed any more.
After about 2 months of letters and many many phone calls we eventually ended up with the pittance they offered and managed to argue keeping the ncb, but only after they received a letter from my solicitor regarding the matter.
The icing on the cake was the letter and fine from the dvla; after all the car was now 'the property of the insurance company' so surely in six months they'd have had plenty of time to sort out the relevant paperwork as they promised they would. Yet more of my time and money sorting out something that insurance company were supposed to have dealt with.
I expected to get what I paid for and for an insurance company to not try and screw me at every turn and evade their side of the bargain. After all, why did I pay them several times the value of the car over a number of years, if not for the eventuality of an accident? It should never have been the case that they baulked at every turn, constantly trying to avoid anything but the barest minimum of responsibility or monetary recompense. As far as I am concerned, the insurance company were just trying it on in order to incur as little expense to themselves as possible...
after they'd taken my money of course.
All in all, what should have been a simple matter, dragged on for months and left me without a vehicle or any money with which to buy another during that time.
Needless to say I changed companies as soon as I was able to.
It may also be of note that the independent assessor garage used by that insurance firm were done for fraud and ringing cars (amongst other vehicle related offences) about 2 years later.
Had my dealings with insurance companies over the years been more positive and less protracted and difficult, perhaps I might feel less aggrieved by the inconvenience of having to have anything whatsoever to do with them. To say I don't like insurance companies and the way they operate, with the law to back them up against you so you have no other choice except to pay out each year or month with the prospect of getting very little in return, is putting it reasonably. My personal feelings towards them are, shall we say, less than reasonable.