Even though it's "Εγώ", which of course is nominative, since there is no real subject of this phrase - no one is doing anything - then I think that "me and my cousin" is fine, since the default case in English is the accusative, not the nominative. This phrase would be translated that way by most native English speakers. "Who's there?/Who's going?" etc, would end up as "me and my cousin", not "my cousin and I".
I think the guiding principle should always be what a competent teacher of English would currently teach. If Duolingo accepts vernacular translations then the number of regional variations could be immense and confusing, especially as it seems many of the followers of this course do not have English as a first language.
"Give us them books", rather than "give me those books" would be a very common verbal construction in parts of the U.K. But I don't think anybody would teach it or accept it as a translation.
Good point and I believe it's been, or should be, our guiding principle. And it needs to be pointed out that this sentence has been deleted and replaced with: "Ο ξάδελφος μου και εγώ" thus promoting the more correct English usage of: "My cousin and I".
We do also accept "My cousin and me".
It all depends on whether the teacher is correct or not, not on whether they're a teacher.
"Give us them books" is wrong, but that's not what we're talking about here. English does not behave as Greek does in this (the sentence being translated) particular case, and it's not wrong to translate the Greek as "me and my brother", just as was never wrong to split an infinitive in English, say, just because you couldn't do so in Latin.