If my opponent has Goodra with the "Slimy Room" Ability in the Active Spot, do I still have to flip a coin to attach an Energy to one of my Pokemon that already has a Fusion Strike Energy card attached to it?

Yes you still have to flip a coin, because the Slimy Room Ability is acting on you the player not on a specific Pokemon; and Fusion Strike Energy's effect applies to the Pokemon it is attached to, therefore it does not block Slimy Room's effect.

Relevant cards

SWSH8_197

SWSH8_244