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.
I agree with J.M. and Jimi.
It's semantically acceptable to use a gerund following a possessive adjective/determiner, but, as with anything, it can be clearer in some sentences (e.g., "his new shoes really helped with his running...") than in others ("Company X experimented with its expanding into a new business area...").
Just my two cents!
Best Answer
To take the last question first: no, the forms in the examples in the question here are not gerunds. They are verbal nouns.
Morphologically speaking, gerunds and verbal nouns are indistinguishable in English: they both end in -ing and are identical both to each other and to the present participle (which is a different kettle of fish that I’ll leave out of this).
The only way to tell them apart is by their syntactic properties (but not, as rogermue surmises in his answer, their syntactic function): whereas a verbal noun is a true noun and functions as a true noun, a gerund has both noun-like properties and verb-like properties:
The two types are in complementary distribution, then.
The grammatical examples in this question are all instances of verbal nouns, rather than gerunds: they are preceded by determinatives (and we can switch the possessive ones for definite articles without losing grammaticality) and adjectives; and they modify their ‘object’ by using of-constructions.
The ungrammatical examples, on the other hand, attempt to shoehorn a verbal noun into being a gerund by having a pronoun precede it, though maintaining the modifying adjectives and the of-constructions. Ungrammaticality naturally ensues.
If we convert the ungrammatical examples into actual gerund clauses by also substituting adverbs for adjectives and direct objects for of-constructions, they become completely grammatical:
Note that, while both options (verbal noun or gerund) are equally grammatical, one is frequently preferable to the other, even in syntactically similar circumstances: the choice depends on a variety of factors, but most basically I would say it is a semantic one. The gerund expresses the simple fact that the verbal action was carried out, whereas the verbal noun tends to express the manner in which it was carried out. So if the manner in which something happens that is important, the verbal noun is preferred; if it’s the simple fact of the action taking place at all that’s relevant, the gerund is preferred; and if either or both is relevant, both work equally well:
As the examples above show, the answer to the first question is never: a gerund that expresses a subject can always do so through an objective pronoun. It is only when dealing with verbal nouns that objective pronouns are not an option.
On the other hand, it is frequently the case that the subject of a gerund can be expressed by an objective pronoun, but not idiomatically by a possessive determinative. In other words, the exact opposite of what the question here asks. In this case, the difference is, as far as I can tell (I’m sure GaGEL deal with this in more detail, but alas I have no access to it), based on what components the gerund clause contains:
In other words:
My gut instinct is to call the last example entirely ungrammatical, but I have a hunch some would consider it acceptable.
This actually implies that there are really three levels in the verb↔︎noun hierarchy of non-finite verbish forms:
(1) thus has no verb-like properties at all, only noun-like properties; (2) has the verb-like property of taking direct objects and optionally also NP subjects; and (3) have virtually no noun-like properties, allowing direct objects and requiring NP subjects, like finite verbs.
1 Note that ‘possessive determiners’ here refers to any possessive expression that can fill the determiner slot in a NP. That includes the possessive determinatives (my/your/his/her/our/their), but also NPs with possessive clitics (Bill’s, the King of Spain’s, etc.).