Does going to the bench reset Kricketune V's "Exciting Stage" Ability?
No. Even if it leaves play and comes back, you can still only use one Exciting Stage Ability per turn even if you have 2 different Kricketune V in play. It is a Game State check, not an effect.
Source: Battle Styles FAQ; TPCi Rules Team (2021-03-18)
Does the box regarding taking prizes for KO'ing a Pokémon-ex from the EX-era (like EX Ruby & Sapphire) count as a "Rule Box" for the purpose of Empoleon V's "Emperor's Eyes" Ability?
No, the term Rule Box does not apply to anything prior to Black & White. As of the date of this ruling these things are covered by Rule Box: Pokémon V, Pokémon VMAX, Pokémon-GX, TAG TEAM, Prism Star, Pokémon-EX, Mega Evolution, and Pokémon BREAK (any future cards that may have a Rule Box are also covered.)
Source: TPCi Rules Team (2021-03-18)
If my Mimikyu V with the "Dummy Doll" Ability protection is moved to or from my Bench during my opponent's turn by a switching action such as Boss's Orders is it no longer protected?
It is still protected. It is an effect from an Ability, not from an attack, and so is not removed by moving to or from the bench.
Source: Battle Styles FAQ; TPCi Rules Team (2021-03-18)
If I play Mimikyu V with the "Dummy Doll" Ability to my Bench and then make it Active, does it keep the protective effect of "Dummy Doll", or is that removed when it moves from the Bench to the Active Spot?
You keep the protective effect of the Ability; it's not like losing an attack effect when moving from the Active Spot to the Bench.
Source: Battle Styles FAQ; TPCi Rules Team (2021-03-18)
If I have Kingdra with the "Deep Sea King" Ability on the bench and my opponent uses Heatmor's "Burning Licks" attack that discards energy cards from my Active Pokemon and also knocks it out, can I move those energy cards to Kingdra before they are discarded?
No, the discard effect of the attack happens first, but you don't move energy for Deep Sea King until after you resolve the Knock Out.
Source: Battle Styles FAQ; TPCi Rules Team (2021-03-18)
My opponent has Galarian Weezing with "Neutralizing Gas" Ability in play. Is the "Top Entry" Ability still allowed as a method of benching a Pokemon, or can it not be benched?
You can still bench it. The Top Entry Ability is used before the Pokemon with that Ability is "in play".
Source: TPCi Rules Team (2021-01-21)
Is a Pokemon put into play via the "Top Entry" Ability deemed Stage 1 regardless, and not a "Basic" Pokemon as they're technically a Stage 1 Evolution?
It retains its Stage, but it is not an evolved Pokemon ("a Pokemon in play that has another Pokemon under it") even though it remains an "Evolution Pokemon" ("a Pokemon card that evolves from a lower stage Pokemon"). Note this difference in terms because it is important in some interactions.
Source: TPCi Rules Team (2021-01-21)
If I play a Pokemon with the "Top Entry" Ability and it goes to the bench as a Stage 1, what happens if I then use Devolution Spray Z?
You can't use Devolution Spray Z on it since although it is an "Evolution Pokemon", it is not Evolved because there is no Basic underneath it. Either you can't play the card, or if it has already been played (it's no longer in the hand) another target will have to be chosen. If there are no other legal targets, then the play has to be reversed.
Source: TPCi Rules Team (2021-01-21)