Is it true that this sentence can be either ''his name is Peter'' or ''her name is Peter''? Let's say Peter could be both man's and woman's name - I could not figure out the gender of the mentioned third party. Right?
Yes. You're correct. It could mean both. You could point to a woman, man or child and replace Peter with that person's name and leave the rest of the sentence as it is, and you would be correct.
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.