Attacks » Copycat
Can I use Mimikyu's "Copycat" attack to copy an attack like "Flare Strike" that says "this Pokemon can't use Flare Strike during your next turn" and then on my next turn use "Copycat" to copy that same attack again?
Attacks like Flare Strike put an effect on the Pokemon that prohibits using that particular attack on your next turn. But Mimikyu's Copycat attack doesn't have that restriction; so yes, you can use Copycat to copy Flare Strike multiple turns in succession.
Source: TPCi Rules Team (2019-11-14)
Attacks » Copycat
When I use Mimikyu's "Copycat" attack, do I need to have the same required energy as the opponent's previous attack?
No, you only need whatever energy is required on Mimikyu in order to use its Copycat attack. However, if the attack requires you to do something with a particular type of energy you don't have, you cannot use that part of the copied attack.
Source: Team Up FAQ (2019-01-18), TPCi Rules Team (2019-01-31)