as a capital fanatic I am not sure if there is any capital on "E" that satisfies all conditions.
There's Edinburg, Scotland, but Scotland is not member of UN
Also, croatian name for capital of Armenia is Erevan, but I believe english version is Yerevan?
However, if I missed something I'll be glad to hear it, if not I believe other one can continue with F
Yes, you're right; in English, the name for the Armenian capital is Yerevan.
Gabarone, the capital of Botswana
Should be P Gus, I've been warned about you.
Paris, France