The answer that Jason Bassford linked has a great explanation of the context in which to use possessive pronouns (eg. my/mine, your(s), his, her(s)) and objective pronouns (eg. me, you, him, her). I slightly disagree with the conclusion that using possessive pronouns before gerunds will "never get you into trouble" since they are usually jarring in spoken sentences, but they are a fairly safe choice for formal written English.
To answer your specific questions, the sentences
I don’t like his insulting me
I don’t like your talking to me
sound old-fashioned and somewhat stilted to me due to the use of possessive pronouns, and it would be unusual to hear this in modern, spoken English. The other sentences sound more natural because the possessive pronoun her happens to be identical to the objective pronoun her. If those sentences had instead used the possessive pronoun hers, they would also sound outdated.
If you have to write research papers or academic essays in English, using possessive pronouns before gerunds is acceptable and perhaps is still the standard form. However, you would not say "I enjoyed your answering my grammar question." in a paper or essay since those forms of writing typically do not use the second person.
The forms of writing where you would use the second person are letters, emails, and internet messages/comments, and in these forms it would be more natural to use an objective pronoun before the gerund, such as "I enjoyed you answering my grammar question." Using objective pronouns before gerunds in spoken English is also more natural-sounding than using possessive pronouns. However, this advice applies more to modern English than to English from even several decades ago.