"Helen drinks milk."
Translation:Η Ελένη πίνει γάλα.
Does this determiner only need to be used when the subject is a proper noun? We have seen proper nouns addressed without it, e.g. Ευχαριστω, Ελενη.
No, it is used both for proper and common nouns, but only if needed. It is not needed in your example, because that's the vocative case and it doesn't take an article!
So, then, a determiner is always needed in front of any subject or anything in the nominative case?
Alas, if only it were so simple! You may have already seen it in the lessons, but you can omit indefinite articles: "Είμαι αγόρι" = "I am a boy" with "a" missing. Come to think of it, you can probably omit just the indefinite articles but not the definite ones?