When does character death happen

It's important to remember relatively few combats are actually with the goal of killing people - the killing or violence is usually a means or a thing in order to get another goal. The three easiest are: Get something (theft, robbery, occupy a location), Defend territory (scare folks off), Ego (thrash them and teach them a lesson, humiliate them, show that you're in charge).

Dying is usually the least interesting result, because it means the problems (for that character, at least) are over - the best results are are "How will losing this combat make your life more difficult in the future?"

I've written my Big List of Combat Stakes which covers a lot of this.

No.

The way damage works is that you take it until you get to 0 HP. On the strike that takes you to 0, you check the remaining damage against your max HP, if it's higher or equal to, you are dead.

If you suffer damage while at 0 HP, three things happen. First you suffer a death save failure. Second, if it's a crit, you suffer a second death saving throw failure (three failures means death). Finally if the damage you take meets or exceeds your max HP, you die.

However, the damage at 0 HP only counts for the single hit you are taking. Thus if you take 10 damage, and the next turn take 15, you are still at 0 HP, and just have two death save failures, but no further consequences.