Do death triggers still happen if I am at 0 or less life after combat damage?
Here's what happens:


Combat damage is assigned and dealt - no stack
Your life total is set to (less than) 0, creatures that were dealt lethal damage are moved to the graveyard. Cutthroat;s ability is placed on the stack X times, once for each creature that died.
Before Cutthroat's triggered ability can resolve however, you have already died because coming to 0 life or less is a state-based effect. Whatever's on the stack doesn't resolve anymore.