If damage that a source would deal to you can't be prevented, you still put an incarnation counter on Nine Lives.

Once the first triggered ability is on the stack, it won't trigger again, even if more damage is prevented and more incarnation counters are put onto it.

If more than one source deals damage to you at once, prevent the damage from each of them and put that many incarnation counters on Nine Lives, even if this causes more than nine incarnation counters to be on Nine Lives. Leave it to cats to abuse loopholes.

The last ability of Nine Lives triggers no matter how it leaves the battlefield.

If you counter the first triggered ability or otherwise remove it from the stack, it will just trigger again immediately if Nine Lives still has nine or more incarnation counters on it.