The points are meaningless in terms of your record, unless for some reason your job cares. Don't get another ticket for the next year from when the ticket is settled, and they will go away on their own.

Insurance looks at the actual violations, not the points. A careless ticket is equiavlent to a small speeding ticket in most cases (however every person is viewed differently by insurance).

If you have been with your company for 7 years, have a clean record otherwise, they won't move your insurance much, if at all. The accident on the other hand is another story.

Your insurance MAY offer you some kind of discount for defensive driving, etc.

As for fighting it, well, you were in an accident. Pretty slam dunk case for careless driving. The cop COULD have potentially hit you with something worse, and there really isn't anything else you can plead down to that would look better for insurance.

You could go to court, and maybe the prosecutor is in a good mood, and they will dismiss it if you pay court costs, but thats a stretch, and the cop would have probably said something along those lines if it was a case of "look we just want to document this".

Edit: Ultimately though, the best thing to do is contact your insurance agent (hopefully you have a local one and just didn't go with a faceless online, whoever is cheapest company, and switch every 6 months). They will have a good sense as to how it may affect your rates, what some options are for you to reduce any hit, and may very well be familiar with the town and can give you some ideas of what can be done on that end.