Translation:They don't have daughters.
"They have no daughters" equals "They don't have daughters" and should be accepted. The main solution is incorrect English, that is "They have not got daughters." It is not slang,, but wrong usage, and therefore, the correct solution should be "They have no daughters" or secondly "They don't have daughters.
Your correction was "They have not got daughters" We wouldn't really ever say this in the UK as it sounds wrong. I wrote "They have no daughters" and I think this should be accepted too
In the US this is also the most common way to say it (at least, in the West)
Why is it children 孩子, then son 儿子, but daughter replaces the 子 and changes the order of 儿 to form 女儿? Is it just like that or it has an underlying sense?
While the sentence doesn't indicate how many, it is more common in English to say and sounds more natural to say "they don't have any daughters", as opposed to "they don't have any daughter".
"They don't have a daughter" would be correct for the singular (as you say the Chinese doesn't indicate how many) The reason "any daughter" doesn't sound natural is because "any" is used in front of nouns to refer to indefinite or unknown quantities; all countable nouns (e.g. apples, chairs, daughters...) are in plural and all uncountable nouns (e.g. money, bread, etc.) are in the "singular".
I translated it as "They have no daughters" and it was marked wrong; and corrected as "They have NOT GOT daughters". Isn't this the same meaning? And is DIFFERENT FROM the translation above!!
"他" means "they", not "I". The sentence "I have no daughters" would be translated as "我没有女儿".