Depends on the context. If everyone knows that the man has a sibling but doesn't know it's a sister or a brother, then you can use «У мужчины сестра» to say the sibling is actually a sister.

«Есть» is used when existence is important. However, if everyone knows he does have a sibling, existence is not important and the important part is «сестра» (i.e. you're not making a statement about the man having someone, you're making a statement about who is the sibling the man has). But if it wasn't previously known if the man has any sibling, then «есть» is required because you're making a statement about existence of a sister.

У мужчи́н есть сестра

If you wrote «У мужчины есть сестры», it wound be understood to mean «У мужчи́ны есть сёстры» 'The man has sisters' because it's allowed to drop the dots over «ё». However, this is not what the English sentence says.

«У мужчи́ны есть сестры́» (with plain «е», not «ё») is ungrammatical: «сестры́» is a genitive case, and this sentence requires nominative, because 'sister' is the grammatical subject (literally it's something like 'at man, [there] is sister).

Here "у мужчины" is the genitive case of a singular "мужчина". It's not plural.

Yes.

«Сестра» does not need to be declined in this sentence; it will remain in its Nominative singular form. Think of this sentence as meaning “There is a sister with/by/of the man.” That’s pretty much what it translates to literally. So, the only word that will need to be declined to Genitive case is «мужчина» (which is spelled «мужчины») because of the preposition «у». Genitive case normally applies to words that are the origin or possessors/owners of other words.

For instance:

“The cat has a mouse.” = «У кошки есть мышь.»

“Cat,” in its Nominative singular form, is «кошка», and Genitive singular is «кошки». The Nominative singular form for “mouse” is «мышь». Notice the word for mouse did not change, but “cat” did, because it is the possessor of the object.

Another example:

“The boy threw the dog’s ball.” = «Мальчик бросил мяч собаки.»

«Мальчик» is in its Nominative singular form because it is the subject, and «мяч» is in accusative singular because it is the direct object of the verb “to throw,” but because “ball” is an inanimate masculine noun, it does not have a different form for this case. And, finally, «собака» (Nominative singular for “dog”) is declined to Genitive case «собаки» because it is the owner of the ball.

No. Because of the preposition «у», «мужчи́на» must be declined to the genitive case, which is «мужчи́ны».

This is a great website for checking declensions: http://starling.rinet.ru/cgi-bin/morphque.cgi?flags=endnnnnp

Человек — это 'person', или мужчина, или женщина.

No. The sister is nominative because it's a grammatical subject of the sentence.

In English, it's object: "The man has a sister", 'the man' does an action of 'having' directed at the sister. So, sister is grammatically an object, she doesn't do the action but an action is directed at her.

However, Russian treats things differently. In Russian, sister is the subject, and sister exists near the man: "By [the] man, there-is [a] sister". (Of course, we don't translate it this way, this is just to show the grammar.) So, here's the sister that does the action of 'being', and she is being by/near/with the man. Sister is grammatically the subject, she is the one who 'is'.

No, it's not apparently wrong. In fact, it's exactly the suggested translation (У мужчины есть сестра). Looks like a bug with the input in Roman alphabet. :/

У твоего мужчины есть сестра.

In Nominative case, yes, but since we are talking about possession and using the preposition «у», it requires Genitive case, which would be singular «мужчины», plural «мужчин».

етого мужчины means "THIS man" where as it seems that 'the' and 'a' are not very separated in russian unless you want to really emphasize it and then I think you, in this case, use етого.