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)

Relevant cards

Note: All applicable cards may not be listed below.
XY3_80