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").
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.)
since it is talking about "no brothers," it shouldn't be plural, but should be singular. Bruder not Brüder.
Eh? That doesn't follow at all.
"no brothers" = keine Brüder. Plural: implies that people often have more than one brother.
"no brother" = keinen Bruder. Singular: implies that people usually only have one brother.
Both singular and plural are possible in negative contexts, but they imply different things.
You would use singular for things usually only have one of (e.g. "I don't have a father any more" or "I don't have a car") but plural for things that people (or whatever object is the "owner") often have many of (e.g. "I don't have any books in my home" or "This house has no windows").
You wouldn't say "I don't have any fathers any more" with plural because most people only have one father. And you wouldn't say "This house has no window" because that implies that you expected the house to have exactly one window, which is unusual for a house.
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"