If I attach a Basic Fighting Energy to Eevee, can I use its "Energy Evolution" Ability even though it's known that there's no Fighting type Pokemon that evolves from Eevee?
No, since it's known that there is no Fighting type Pokemon that evolves from Eevee, the Ability cannot be used.
Source: Furious Fists FAQ (2014-08-28), TPCi Rules Team (2014-08-28), TPCi Rules Team (2023-06-22)
For Eevee, can you attach an energy and use its "Energy Evolution" ability and evolve on your first turn? How about on Eevee's first turn in play?
Yes. You can evolve on your first turn and also on the turn that you played Eevee via this Ability.
Source: Furious Fists FAQ (2014-08-28), TPCi Rules Team (2014-08-28)
If I attach a Rainbow Energy to Eevee, what Energy type does it count as for "Energy Evolution"?
Rainbow Energy is not a Basic Energy card. It will not trigger the Ability.
Source: Furious Fists FAQ (2014-08-28), TPCi Rules Team (2014-08-28)
Say I attach a Water Energy to Eevee and do "Energy Evolution". I want to find a Glaceon but unfortunately I only have a Vaporeon in my deck. Do I have to attach the Vaporeon to Eevee?
No. Any search for a specific card or type of card in your deck can be "failed" at your option.
Source: Furious Fists FAQ (2014-08-28), TPCi Rules Team (2014-08-28)
Will Eevee's "Energy Evolution" ability still trigger if it has an Energy attached to it while it's affected by a Special Condition?
There is no text on the card to say it's shut off, so it still works even when afflicted by a Special Condition.
Source: Furious Fists FAQ (2014-08-28), TPCi Rules Team (2014-08-28)
If Eevee only has 10 HP left and Frozen City is in play, would it be KO'd before I have a chance to use its "Energy Evolution" ability by attaching a Basic Energy?
No, it would not. All triggered effects have to be resolved before checking for KO's, so you would take the +10 damage for Frozen City first, then use Energy Evolution to search your deck. Of course, if you fail the search and Eevee's damage is greater than or equal to its HP, then Eevee is KO'd.
Source: TPCi Rules Team (2014-08-28)
If I attach a Basic Energy to Eevee and trigger its "Energy Evolution" ability but choose an evolution belonging to Team Plasma, would I take the damage from Frozen City stadium or not?
The mandatory effect of the Frozen City stadium comes before the optional effect of Energy Evolution. If the Eevee is not a Team Plasma Pokemon before choosing whether to use Energy Evolution, you would take the damage from Frozen City.
Source: TPCi Rules Team (2014-08-28)