Can I use Rare Candy on a Scatterbug the first turn of the game or the turn that Scatterbug is played, since "Adaptive Evolution" says it can evolve in those cases?
No, you can't use Rare Candy to evolve in either of those conditions because the text on Rare Candy specifically prevents it.
Source: Scarlet & Violet FAQ; TPCi Rules Team (2023-03-30)
Can I use Evosoda on a Caterpie the first turn of the game or the turn that Caterpie is played since "Adaptive Evolution" says it can evolve in those cases?
No, you can't use Evosoda to evolve in either of those conditions because the text on Evosoda specifically prevents it.
Source: TPCi Rules Team (2014-05-29)
Can I use Caterpie's "Adaptive Evolution" Ability to evolve into Metapod, then immediately use Metapod's "Adaptive Evolution" Ability to evolve into Butterfree all on the same turn?
Yes, you can.
Source: Flashfire FAQ (2014-05-08), TPCi Rules Team (2014-05-08)