No we do not die, and of course brainwave activity does not stop either, as rem sleep shows during dreaming. Now, want to give us a whirl on how dreaming can occur without brainwave activity? ... Yeah, I didn’t think so.
If you'd read my previous posts, you know that there is brain activity in at least some cases, as people who have undergone this procedure have told the doctors about what went on in the operating room while they were "out". Apparently, the equipment they use to measure brain activity in these scenarios is either not working properly, or there can be brain activity that is at too low a level to be measured.
And yes, if a person can be aware of what is going on in his/her surroundings, then a person can also have enough brain activity to dream.
Further, there are numerous reports of hallucinations that have occurred during these procedures. Some are religious in nature, some are not.
Do the research yourself if you like. It wasn't all that hard.