If I use Charizard's "Roaring Resolve" Ability with only 10hp left, does he get KO'd when I place the 2 damage counters before or after I search & attach 2 {R} Energy cards?
You must finish all the text of the Ability before checking for knockouts, so yes you would search your deck and attach {R} Energy cards before KO'ing Charizard.
Source:
Team Up FAQ (2019-01-18), TPCi Rules Team (2019-01-31)
Say Ditto uses its "Transform" Ability to become a Klefki, then Klefki uses its "Wonder Lock" Ability to attach it to another Pokemon as a tool. When doing this what happens to the Ditto and Klefki cards?
The Ditto card gets discarded when Klefki is attached as a tool.
If Klefki with the "Wonder Lock" Ability is attached to a Pokemon as a Tool card, what happens if "Silent Lab" stadium comes into play? Does "Wonder Lock" shut off and Klefki get discarded, or does it remain attached to the Pokemon?
Once Klefki is attached as a Pokemon Tool card it retains its effect even when its Ability has been disabled. Once it's attached, it is essentially treated as a Tool with that text, so shutting off its Ability doesn't matter at that point. However, if something such as Lysandre Labs is in effect, Klefki won't be discarded at the end of your opponent's turn.
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.
Say my opponent has a Sudowoodo with the "Roadblock" Ability on his bench, but it's currently disabled due to my active Alolan Muk's "Power of Alchemy" Ability. If I have five Pokemon on my bench when Alolan Muk gets KO'd and "Roadblock" becomes enabled, do I have to discard one of my five benched Pokemon before promoting a new Active Pokemon or not?
When you remove your Active Pokemon after resolving all effects that caused it to be KO'd, you must replace it immediately before resolving any other new effects that take place.
If I use Sylveon's "Wink Wink" attack to have my opponent discard Ilima, can I use Victini's "Victory Star" Ability to force my opponent to re-flip their coins?
Yes, you can. If your opponent flips a coin as part of your Pokemon's attack, you can use "Victory Star" to force them to re-flip those coins; but keep in mind you will have to re-flip your coins for Ilima too.
How does Slaking's "Lazy" Ability or any other form of Ability denial that only applies to Pokemon "in play" interact with Abilities like Giratina's "Distortion Door" Ability that puts a Pokemon card from the discard pile onto your bench?
Since Distortion Door happens when Giratina is not in play, you can use it to put Giratina onto the bench regardless of Lazy or other similar Abilities. You also get to put the damage counters onto your opponent's benched Pokemon too, because you must complete the action of Distortion Door prior to it being shutoff due to Lazy.