When Ezio’s triggered ability resolves, it checks to see if that player has 10 or less life at that time. If they do, you may pay {W}{U}{B}{R}{G}. A second “reflexive” ability triggers when you pay this cost. Each player may respond to that triggered ability as normal, but it doesn’t matter what happens to the damaged player’s life total at that point; when that reflexive triggered ability resolves, that player will lose the game no matter what their life total is.

Once an Assassin creature or commander you control deals combat damage to a player in a turn, you can cast spells for their freerunning costs for the rest of that turn. It doesn’t matter what happens to that Assassin or commander after that.