Him and I, Her and I should be acceptable considering that dia also means him or her
Do we have a problem with English hypercorrectivity here? Him and me should also be fine. As we have no context there is no way of knowing if it should be subject or object first person.
Dia means both she and he, so both 'he and I' and 'her and I' should be correct answers
Mixing a pronoun (I) with an object (him) is bad English. You wouldn’t say ‘Him and I went to the zoo’. And for the purposes of good use of language, Duolingo rightly isn’t encouraging people to say ‘Me and him went to the zoo.’ It may be increasingly common but it’s incorrect.