Yes, it should be accepted. If it’s not, please use the 'Report a Problem' button.
«Нет» is not just 'no', it's also used to express absence, like 'there is no'. So, «его нет» literally means 'there is no him'. This is not something we would say in English, so that's why it needs to be re-phrased as 'he is away'.
a) — Он дома?
— Нет, его нет [дома].
б) — Она делает домашние задания?
— К сожалению, нет. / К сожалению, не делает.
No, that would be «Извините, не он» (or, depending on the context, it might be another case).
I also translated this as "I'm sorry, he is not,"* but not as a existential thing. It's a common wording for correcting an assertion someone has made about a third person:
"Ivan Ivanovich is the eighth winner of the Such-&-Such Award."
"I'm sorry, he is not. The Awards were canceled last year and there was no winner, so that makes Ivan Ivanovich only the seventh winner of the award."
(*I was marked wrong for this on 13 August 2018, and have reported it.)
I’m afraid this is not a possible translation.
It might teoretically work if you use nominative («он — нет»), but even that doesn’t really sound natural in your context. I’d say «Извините, это не так» ‘Sorry, it’s not true’ or something.
Он is nominative case. You need to use the genitive form, which is его (look it up on a declension table of pronouns).
Genitive is necessary because of нет. The word order is backwards from English - if if the word order were tuned to English, then it would be нет его, and you have to use genitive for words negated by нет.
I got this as a listen and write. I write it perfectly but it made no sense. "Его нет" means "He is away"? Really? Perhaps in a specific context but I read it as "He isn't" which has a much more broad application for learners. Is he hungry? Is he there? Is he good? Is he sleeping? etc. Can его нет be used for these questions too?