If Haxorus uses its "Bring Down the Axe" attack while Temple of Sinnoh is in play and there is a Mist Energy attached to the Defending Pokemon, does it knock out the Defending Pokemon?
Yes. Even though Temple of Sinnoh turns off the text portion of Mist Energy, it still remains a Special Energy Card and meets the requirement for Bring Down the Axe.
Source: TPCi Rules Team (2024-09-12)
If my Grimmsnarl uses its "Goad 'n' Grab" attack and my opponent's Active Pokemon has a Mist Energy attached the switch out effect will be prevented, but will it still do 160 to the unchanged Active Pokemon?
No, it will not do any damage because there isn't a NEW Active Pokemon.
Source: Stellar Crown FAQ; TPCi Rules Team (2024-09-12)
If my opponent used Iron Boulder ex's "Repulsor Axe" attack, will my Attacking Pokemon with Mist Energy attached receive the 8 damage counters from "Repulsor Axe" if I attack and do damage to Iron Boulder ex?
No, it will not receive the 8 damage counters. Mist Energy prevents all effects of your opponent's attacks done to the Pokemon where it is attached, and Repulsor Axe's damage counters are an effect of your opponent's attack.
Source: Temporal Forces FAQ; TPCi Rules Team (2024-03-21)