Generally anything negative on your record such as speeding will hurt your chances. Points are similar, but also vary state by state. In California, you would have only been able to remove one of those points through traffic school, the other two you would not be able to remove. They would "fall off" your record after three years, but companies here usually ask for a ten (10) year record so they would still see it, but I don't believe (don't quote me on this) that it would count against you after those three years insurance wise. A lot of companies here policy is either one point or one accident (not one point and one accident), or up to two points at some. So here you'd qualify for some companies, but not all. When it comes to hiring, I think your points/driving record would come into consideration.
18 months isn't that long, and I personally don't believe you've paid your dues. How can you be trusted to drive an ambulance? My number one pet peeve at work is people who say "we're an ambulance" when they speed, park in the red zone, possibly play with the lights and sirens at inappropriate times. With your record, I wouldn't trust you driving an ambulance because I'd be afraid that you'd have the same attitude especially since you speed even when you're not in an ambulance.
Also search the forum. This is a regularly asked questions. I'm not just saying this because you created a post, but I want you to see previous feedback to other people who've asked the same question.