"No, I do not have a family."
depends. は emphasizes that 'kazoku' is being talked about "About family, there isn't one". Where as が is more of a mention that "family isn't". i like to think that が is answering a silent question so think of it as being the response to "What don't you have?", with the answer being かぞくがいません。(I don't have a family) where as は is focusing more on the word family so more of a reply to "Where is your family/ Don't you have a family" and therefore かぞくはいません。(I don't have a family).
Yes おります is a common expression when speaking to clients or bosses.
I think the algorithm of the correct answer suggestion is to find a correct answer in the word bank and provide the one with the least difference, in this case ありません=>おりません(1 difference) over いません(2 differences). I am not an engineer of Duolingo so I cannot tell if suggesting いません is possible or not.
Duo doesn't grade punctuation so the comma is optional,
Duo does grade spaces though and will mark you incorrect if you add a space to the Japanese where none exists.
The Japanese comma is a full kana width 「、」so there is no use of a space after like there would be in English.
「いいえ、家族はいません」comma with no space and「いいえ家族はいません」no comma and no space, are both acceptable.
Think of ではありません as a set construction for saying "Is not"
家族ではありません would be "[It/They] is/are not a family", the opposite of 家族です "[It/they] is/are a family" (です itself is from であります)
で is a form of copula だ used to mark a state of being; it turns the entire phrase before it into a noun phrase, which is then marked with ありません the negative form of the existence verb ある, so it is like saying "In the state of being X, it does not exist"
家族はいません "There is not a family (I do not have a family)" is the opposite of 家族がいます "There is a family (I have a family)"
You have a wrong punctuation mark "," and a space after the ",".
The correct one is "、" and there can be no space between this mark and the next character i.e. "家".
Even if you don't have the punctuation mark, Duolingo should mark your answer as correct. However, the space is the problematic one.