Abilities » Freezing Gaze | Abilities » Lost Out
If my Tyranitar GX knocks out an opponent's Glaceon GX with the "Freezing Gaze" Ability, does Tyranitar GX's "Lost Out" Ability turn on in time for it to send Glaceon GX to the Lost Zone?
Nope, Freezing Gaze is still active when Glaceon GX is KO'd, so Tyranitar GX does not get to use its Lost Out Ability.
Source: TPCi Rules Team (2018-12-27)
Abilities » Lost Out
If you Knock Out a Pokemon using one of Tyranitar GX's attacks, do you get to take a prize card if the Pokemon is sent to the Lost Zone due to Tyranitar GX's "Lost Out" Ability?
Yes, you do. It is still Knocked Out.
Source: Lost Thunder FAQ (2018-10-20), TPCi Rules Team (2018-11-01)