If your question is whether 'The sisters have no brothers' should be accepted: yes, it should.
Ususally you need determiner after "have(n't)" if it's followed by a noun. "do not have any[pl.]/a[sing.] brother(s)" = "haven't any[pl.]/a*[sing.] brother(s)" = "have no** brother(s)".
Without a determiner, most of the time a verb will be expected, e.g. auxiliary + past participle: "haven't been there", "haven't eaten yet" (never *"do not have been"), or auxiliary + infinitive: "don't have to go", "don't have to eat", (never *"haven't to go").
No haven't is a contraction of have not. Ole English doesn't count here. You need "have no".
The "u" in Bruder singular sounds almost like an "oh" sound. The "u" in Brüder plural is very different - the best way I was ever taught this is to put your mouth in the shape you would usually make to say the letter "u" but then say an "e" instead, but keeping your mouth in the "u" position.
It has to be keine here because it's talking about brothers in the plural. It doesn't matter what the original gender of the noun is - as soon as it's a plural, that overrides the original gender.
(If it was just about one brother, it still wouldn't be "kein" though. It would be "keinen Bruder" because it's in the accusative.)
So an "eine (a)" can be followed by a plural? Which here is "k+eine Brüder". Or it is just not proper to use the word "nicht" here?
No, "eine" cannot be followed by a plural but "keine" is about like "n+one". Yes, you could use the word "nicht" here but this can negate almost every other word in the sentence. The word "keine" shows immediately that it refers to the following noun or to its numeral. For example: "Er hat keine vier Schwestern" means: He has less than four sisters. If he has more than four you can say: "Er hat nicht vier, sondern (but) fünf Schwestern"
So "The sisters have not got any brothers." is right and "The sisters have not any brothers"is wrong? Why?
An example here why "have not any" does not fit here: "Have got" -means you possess something whereas only "have not" is not grammatically correct - you would need to use: The Sisters DO not have any brothers.
Duolingo usually accepts my German typing if I forget an accent, and I've had "Bruder" accepted as plural with a little note to remember the accents. But this time, I was penalised. Slightly annoyed at the lack of consistency.
"Die Schwestern" is plural. You could write "The sisters don't have brothers", but to use "doesn't have" you would need one sister (singular): "The sister doesn't have brothers" which is not what the sentence says. I hope that helps!