If I have 5 Pokemon on my Bench and Lunatone with "New Moon" Ability and Solrock are in play, what happens if my opponent plays Collapsed Stadium? Would I have to discard any Benched Pokemon or does "New Moon" prevent that?
The effect of Collapsed Stadium is being done to the player's bench size, not to the Pokemon on the bench; so New Moon would not protect them and you would have to discard Benched Pokemon.
Source: TPCi Rules Team (2023-09-07)
Would a Solrock coming into play on the bench receive damage counters from Gapejaw Bog if a Lunatone with "New Moon" Ability is in play but the Ability is not active yet (it requires Solrock to also be in play)? Or what if you put that Lunatone onto the bench while Solrock is already in play?
New Moon activates immediately when it's condition is met, so in either case the 2 damage counters from Gapejaw Bog would be prevented.
Source: TPCi Rules Team (2023-09-07)
If Lost City stadium is in play and Lunatone's "New Moon" Ability is active, what happens if multiple Pokemon including Lunatone & Solrock are KO'd?
All KO's are evaluated simultaneously, so if the New Moon ability was active all KO'd Pokemon would go to the discard pile.
Source: TPCi Rules Team (2023-09-07)
If Pokemon League Headquarters is in play and I have Lunatone with the "New Moon" Ability and also Solrock in play, do my Basic Pokemon have an extra {C} cost to their Attacks?
No, but your opponent's Pokemon do.
Source: Obsidian Flames FAQ; TPCi Rules Team (2023-08-10)