The Navy tries to be pretty accommodating when it comes to airsickness. If you get sick the IPs have to report it and you'll have to see the flight doc, but they do give you a chance to get over it and continue in the training. I haven't personally seen anyone get the boot for airsickness but I'm sure it has happened.
It happens, but not often. Either you will get over it or you won't.
First you will get meds. If you can fly and not get sick off the meds after a few flights, then you are done.
If the meds don't work or you can't adapt, you will go to the chair and spin there for a while. This cures or at least helps 90% of everyone. If that doesn't work and you just can't make it through C4204 without puking, you will get NPQd.
Guys who puke after 4401 (first solo) are generally just watched closely. You can't puke on a safe for solo flight. Other than that, as long as you can make it through most flights without it effecting you, you will be fine. Regardless, if you are a puker, the squadron will watch you closely and make you a priority to fly so that you can stay acclimated. All IPs are fairly sympathetic and try to do what they can so that guys can make it through a flight if they are having trouble (ie - do the spin last instead of first, only do a few PAs on a PA flight as long as they have met MIF previously, hit the landing pattern first, etc.)
I have had guys puke in every stage, including RI's.