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.

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