To be clear, you cannot die while playing as Godzilla--you'll always turn back into a normal player at the end of it, no matter what. From there, you'll play the rest of the match out as usual. You don't, by the way, get any rewards for having been Godzilla in a match, aside from the happy memories.

The second quest in the Godzilla tab requires you to damage opponents from one story above or higher, which can be a bit tricky since the game doesn’t explain exactly what this