Well, yes you could be ovulating again. You aren't protected against pregnancy.
Nausea is common with pills for some people, and if you really can't stand it, I'd try switching to a different pill and see if that helps. It may take a few months for that side effect to wear off, so you may have to stick it out. However, if you are throwing up after you take your pill, it is not effective. So you should probably switch.
Your period may be really weird from starting and stopping the pill, so don't freak out if it is doing weird stuff.