Abilities » Energy Evolution | Trainers » Frozen City
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)
Abilities » Energy Evolution | Trainers » Frozen City
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)
Trainers » Frozen City
How does the stadium card Frozen City interact when a player attaches Double Colorless Energy to a non-Team Plasma Pokemon?
You only place 2 damage counters because there was just one attachment, even though the attachment provided multiple energies.
Source: TPCi Rules Team (2013-05-16)