"Your friend is my enemy."
Translation:Dein Freund ist mein Feind.
Deine Freundin ist meine Feindin is accepted.
If you're going to be specific about your friend being female, the list of alternatives expects you to specify "female enemy" as well.
dein Freund and mein Feind are both in the nominative case.
If you saw Ihr Freund -- that's the form used when speaking politely to one or more people; Ihr (always capitalised) is the possessive corresponding to Sie (always capitalised), the polite "you".
I don't understand why Gegner isn't usable here with they both mean approximately the same thing.
"to be" is not a transitive verb that takes a direct object.
You can't say that the enemy "is being been" by your friend.
"to be" is a copula - a linking verb that links a subject to a predicate that says something about the subject.
Such predicates are (almost always) in the nominative case in English. (And, traditionally, in English -- conservative speakers will say "it is I" and not "it's me".)