No, you can't put in an Italian word for every English word.
Would it be incorrect if you said, "Aspetto per la mia amica"?
Not quite! The "for" is included in the word aspettare, so saying "aspetto per la mia amica" would be saying "I'm waiting for for my friend"
I think it would.
I await my lady friend should be okay, but it's not accepted. Even "girl friend" with a space should be accepted if you ask me (ladies call their friends this, and would expect to say the same for amica).
I did report that "Girlfriend" was marked wrong. It is very, very common to use lady friend, girl friend, or girlfriend in regards to a female friend.
I wait seems really awkward. Shouldn't it be I am waiting?
I agree: awkward & abrupt. English would normally use the present continuous tense - am/is/are waiting - for the verb to wait.
This could also translate to: I "expect" my friend. Duo did not accept it.
I touched the girlfriend button in my translation and it was wrong, although girlfriend is also a suggested tanslation of amica...
It marked "Aspetto la mi'amica" as wrong. It's clearly how she pronounces it, and grammatically correct, no?