"Er hat keine Mutter."
Translation:He does not have a mother.
Yes, that is more important in German than (U.K.) English. While I have heard people in the UK say "My Mum has not been well" to an employer, you could never say "Meiner Mama ging es nicht gut" in Germany on a professional level / employer - you must say "Mutter" in formal occasions.
Yes, it is because Mutter is feminine and because it is in the accusative case.
See this link for "ein" word forms:
Hope that helps!