When is a gerund supposed to be preceded by a possessive adjective/determiner?

I assume that the following sentences are grammatically correct:

• He resents your being more popular than he is.
• Most of the members paid their dues without my asking them.
• They objected to the youngest girl's being given the command position.
• What do you think about his buying such an expensive car?
• We were all sorry about Jane's losing her parents like that.

I'm still getting used to this possessive gerund structure. It sounded so weird to me at first.

Is the structure used in both formal and informal contexts? Are there any alternative structures that result in the same meaning and are more frequently used?

(Examples taken from http://grammartips.homestead.com/possessivewithgerund.html)

When I first heard about this usage in a grammar lesson in middle school, it sounded weird to me, too. As in the linked page in your answer, my teacher taught us that using possessive pronouns (also known as genitives) is the only grammatical way to mark subjects of gerund clauses. While that way is more traditional and formal, using object pronouns (accusatives) is also quite common.

In chapter 14, section 4.3, of the Cambridge Grammar of the English Language, entitled “Non-finite and verbless clauses”, the main thrust lays waste to the traditional distinction between gerund clauses and present participle clauses, by arguing they all belong to a single inflectional category; namely, gerund-participles. However, there is a paragraph explaining the use of genitives with gerunds:

There is one respect in which ‘gerund’ and ‘present participle’ clauses differ in their internal form: with ‘gerunds’ the subject may take genitive case, with plain or accusative case a less formal alternant, but with ‘present participles’ the genitive is impossible and pronouns with a nominative–accusative contrast appear in nominative case, with accusative an alternant restricted to informal style. Compare then:
[39] i. She resented his/him/*he being invited to open the debate.
       ii. We appointed Max, he/him/*his being much the best qualified of the candidates.

In other words, gerunds (as in example 39i) can take either the genitive (his) or the accusative (him) as subject, with genitive being more formal and accusative less formal. The nominative (he) is not possible as the subject of a gerund.

In participial clauses with a subject (as in example 39ii), there is a similar situation: both the nominative (he) and accusative (him) are possible, again with accusative being less formal, but the genitive (his) is not possible.

The page of “grammar tips” linked in the question confuses informal style with incorrect grammar, a common problem in grammar advice. The versions of the examples with accusative instead of genitive (e.g. What do you think about him buying such an expensive car?) are perfectly grammatical and simply a less stuffy style.

You will find many examples of gerunds with accusative subject—even in formal academic writing—so you should feel free to use whichever of the two formulations seems natural.

It is perhaps worth adding the contrast identified in the ‘Longman Student Grammar of Spoken and Written English’:

When the possessive alternative is used, it focuses attention on the action described in the ‘-ing’ clause. In contrast the regular Noun Phrase form puts more emphasis on the person doing the action.

Just to comment on common usage (in British English, at least):

Examples 1, 2 and 4, which use possessive pronouns, look OK, but are somewhat formal. I'd be more likely to use the accusative forms, namely:

He resents you being more popular than he is.

Most of the members paid their dues without me asking them.

What do you think about him buying such an expensive car?

I can believe that examples 3 and 5, which use nouns, may be grammatically correct, but they look wrong, and I do not recall seeing or hearing that particular construction used. I would drop the "-'s" in both cases.