As previous posts say, its up to the employer.
Peronal insurance companys ask for all prevous convictions and accident in the last 5 years as you will know.
But commercial insurance can be a bit different, depending on the policy the employer has.
I was running 12 vans about 6 years ago, all the vans were insured for any driver, drivers werent named, had loads of different ways of taking out my policys, ours were any driver over 25 years old,( but believe it or not your actualy discriminating people now if you state that in a job vacancy, even though it might put thousands of pounds onto the insurance if you dont state this)
Also all our drivers were allowed 3 penalty point on their licence when starting with us, but once they were started we could keep them even with 9 points on their licence. which basicaly ment we could keep them till they were banned as the next 3 points would ban them.
Most companys will ask you to hand your licence in every 6 month to check they are ok also, I did anyway, lots of drivers will get a duplicate and use that for any points they get, keep the origonal for handing into the company so they keep there job, but if they have a bump they are f"""d.
But if you can get any points taken off I personaly would, but by the sounds of things you got all the points pretty close to each other, so it might be worth while waiting till they are all due off and get it all done at once, costs you every time you send licence away to get points taken off,
Dales