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.
The gerund phrase is the object of the peposition for, not a subject.
A gerund can be either timeless (it can refer to any time in past, present, or future) or refer to a specific time frame. In your example, the timeless gerund leaving the window open works well enough; in who is responsible for leaving the window open, it refers to leaving the window open in general, without referring to a specific time per se.
It is the context, however, that supplies the temporal information: because you are pointing at a window that is now open and was therefore left open in the past by someone, it is clear that you are talking about a "leaving open" that happened in the past. This information is not (and need not be) included in the gerundial phrase itself.
Suppose you, as a teacher, heard a phone ring incessantly in class. You could ask this, while it was still ringing:
Who is responsible for making this noise?
It would refer to the present, but that can only be known because of the context, not because of the gerund itself.
The same applies to infinitives:
I hope to see you soon.
Context points to the future.
I like to kiss horses.
Context suggests to a habit that happens in past, present, and future.
He ceases to impress me.
Context makes it refer to the past.
In some situations, it is desirable or even necessary to explicitly show that the action described by a gerund or infinitive took place in the past; in that case, having done or to have done is used.
Her losing her mind is a terrible thing to observe.
Her having lost her mind is a terrible thing to observe.
Without having, the natural interpretation of the gerundial phrase is for it to be simultaneous: someone is observing the process of her losing her mind as it is going on. This is not necessarily the only interpretation; but, when context allows it, the default is an interpretation simultaneous with the main verb (is a terrible thing to observe). If you add having, you force a perfect interpretation, where the action happened in the past before the main verb; it is already finished and you are observing the result.
In a different context, the same timeless gerund can refer to the past without the need for having:
Her losing her mind is the most terrible thing that happened this year.
The losing has clearly ended, it happened in the past and is no longer going on, and yet no having is needed. You could say her having lost her mind here, just as in your example about the window, but it is not necessary.
Best Answer
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!