Is there a grammatical reason as to why:
“האם הם עושים את זה או האם אנחנו עושים את זה?“
was not accepted, or is it correct? In case it is correct, does it sound weird or overly posh?
No, from my understand ha-im is usually used when the sentence could be confused as to whether it is a question or a statement. For example, do you like chocolate and you like chocolate are equivalent in Hebrew but ha-im can be used to distinguish the question from the statement.
2018 May, there's no ha'im (question marker) or "anu".
(This is for people in the app who are confused why people are discussing things that aren't part of the exercise. It was there, now it's a different example.