Yes, unlike in English, third-person singular nouns in Hindi are gender neutral. In the case of possessive pronouns like उसका, this means that the gender of the possessor cannot be determined from the pronoun alone, only the context in which its used can give any indication of gender. It should be noted however, that the pronoun must agree in gender with the thing being possessed. In this case, that would be नाम (name) which is a masculine noun.
Name's is not name is in "standardized English" (or at least its closest form), although you might say it that way in many dialects.
It is actually the possessive form of name:
(the) Name - one name
(the) Names - many names
the Name's - of the name
the Names' - of the names
That of corse, does not mean that it is wrong to say, but unless you quote somebody you should avoid writing it!
Yes, as a Transgender person, but think about it like this: For Duolingo their are many people.
And that is actually good so. People, who learn other languages, should learn to know the difference between third person singular and third person plural. It is already difficult to explain some people the difference between you as one person (thou) and you as many people (You all).