Does Mega Gengar EX have to have the correct types of Energy attached to it to do an attack that is copied by "Phantom Gate"?
No, it just needs the Energy required by Phantom Gate.
Source: Phantom Forces FAQ (2014-11-13), TPCi Rules Team (2014-11-13)

Relevant cards

Note: All applicable cards may not be listed below.
XY4_121
XY4_35