The English would be more natural as "The man's chair".
Yes! I was thinking that maybe they were worried that this is less precise, but it isn't; it's ‘a man's chair’ that's ambiguous (between the chair of a man and a chair of a man, while ‘one of the man's chairs’ is used for a chair of the man).
Why is is necessary to add "người" ?
Why is it here not necessary to put 'ấy' at the end?
Your Vietnamese is very well. A paragraph or in talking we usually put "ấy" at the end. I think in this situation they didn't use "ấy" to simpler wither learners.