I don't think the vaccine is showing the number of deaths yet, it'll probably be another couple of months before vaccines have a significant effect on deaths.
Of course it is. The first vaccine was given in the US on Dec 15, effective ~10 days later by Dec 25. That's high risk people removed from the pool of potential dead, 45 days ago. Again, ~2m NH residents account for 1/3 of
current deaths, those are being removed by vaccines from Dec as we speak.
I'm not sure that math makes sense. People can be as reckless as they want if the prevalence is super low.
The prevalence won't go down. If you vaccinate people with ~60% efficacy, and in turn they spread it 2x as much by being less cautious, the prevalence will go up.
We know the natural R is around 3-4, and with the current precautions it hovers around 0.8-1.5.