Really you have no grounds for an appeal as you did commit the offence, you admitted it when you paid the fine. The points are spent after 3 years. However, 3 points for this particular offence will not affect your insurance quote, thats just down to the insurance group of the car you want and the fact that you don't hold a full licence for it yet. The insurance normally ignores 1 offense even speeding for insurance purposes. After you pass your test you will, as you say, have 3 points on your licence until Jan 2014, but you cannot get any more points unless you commit another offence. This is surely down to you.
As far as the accident goes, if you fail to declare it, have an accident and the insurer finds out about the accident they can refuse to pay on the grounds that you failed to declare something material. If you are caught without insurance (or invalid insurance because you failed to declare) its 6 points on you licence anyway. The insurance form usually asks whether you have had any accidents in the previous 5 years irrespective of fault.
You are going to have to either pay the rate, or get a smaller engined (lower group) car.