Yes on both counts.
It’s an out-of-turn incidental, and so can be presumed to allow the activation of the ability before the effect of a given critical injury is applied, though this is open to interpretation and a GM could rule that it does not since it does not state explicitly that it can.
“Death” is a critical injury, so if you’re ignoring the effects you’re ignoring the crit. And then since there is no difficulty limitation on Too Tough to Die, there is no reason why it would not apply as well.
To further read into intent, they literally named an upgrade “Too Tough to Die,” so I think they intended the PC(s) in question to be able to cheat death.