If I use Single Strike Style Mustard to put Duraludon VMAX directly onto my bench, can I later use Cheryl to remove damage counters from it despite it not having evolved into the VMAX?
Yes, it is still an "Evolution Pokemon" since it has an Evolution box printed on it. It would not be an "Evolved Pokemon" but that is not what Cheryl requires.
If Temple of Sinnoh is in play, does Meloetta's "Melodious Echo" attack still do 70 damage for each Fusion Strike Energy attached?
Yes. Temple of Sinnoh removes all effects and provides 1 Colorless energy, but it does not change the card name to be something other than "Fusion Strike Energy". So Melodious Echo still does 70 damage for each attached Fusion Strike Energy you have in play.
If I have a Kecleon with the "Chromashift" Ability in play and two basic Water Energy and a basic Fighting Energy attached, and my opponent has a Jolteon with the "Thunderous Awakening" Ability and a Memory Capsule attached, would Kecleon still be considered a Fighting type Pokemon?
No, it would not. Thunderous Awakening would turn off Kecleon's Chromashift Ability completely, therefore Kecleon would be a Colorless type.
If I play "Canceling Cologne" and then somehow switch out the opponent's Active Pokemon, does the new Active Pokemon have its Abilities or not?
The effect of Canceling Cologne is on the opponent's Active Pokemon position rather than on the Pokemon itself. So if the opponent's Active Pokemon changes after Canceling Cologne has been played, the new Active Pokemon would have its Ability removed and the previous Pokemon would have its Ability restored.
Source:
Astral Radiance FAQ; TPCi Rules Team (2022-05-26)
If I play Phoebe while my opponent has a Dunsparce with "Mysterious Nest" Ability in play, do my Pokemon VMAX's attacks calculate Weakness if applicable or not?
The term "any effects" does not include effects that alter Weakness and Resistance. So Phoebe does not allow you to ignore Mysterious Nest's effect.
If you use Peonia and put a Treasure Energy or Dream Ball from your prizes into your hand, can you use the Treasure Energy or Dream Ball effect to attach it to one of your Pokemon?
Sorry, but Treasure Energy and Dream Ball only trigger when you are taking it as a face-down prize card, not when you are trading it with another card.
Source:
TPCi Rules Team (2021-10-07); TPCi Rules Team (2022-04-07)