LOL! I never go anywhere without a stack of rebirth scrolls...
And no, the game doesn't keep track of how many times you've died.
BTW, neither Lady Tigra nor Stripes doesn't consider Tiger such a bad name after all...
Hmmm... Tiger... Babar... Ratzy... Narinia... Narinia?
One thing that may factor in... is Tiger in the first slot? Party damage (like walking into a fire) affects the first slot, while crush damage (doors, elevators, etc.) affects the last slot.
And yes, you do need to charge the Staff of Death in the Black Fire for it to affect Cet. Once it does, though... swing away!