"I'm home" and "Are you home?" are fine in BrE and, as far as I know, AmE. They are not always exactly interchangeable with the versions with "at". "I'm home" is often used to announce that one has arrived home. "Are you home?" tends to have a nuance of asking whether someone has arrived back home after being out or away.
I don't think that "work home" is correct in AmE, but perhap