Learn English – Ambiguity when a sentence contains multiple possessive pronouns

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.

All of your examples have something in common: though some of the attributives that you give have the form of noun phrases, they are all primarily adverbial in usage:

• Today we're watching the news.
• This week we have to make a schedule.
• Tomorrow we should have lunch.
• This year they gave a good report.
• We visited there last summer.
• She had the records of the meetings last year.

These adverbial phrases have limited currency as nouns. You can sometimes use them as the objects of prepositions, as in Before yesterday I had never heard of her, or as the subject of a sentence as in Today was a good day. However, they don't show the full range of syntactic variation that ordinary noun phrases do: they can't be pluralized and they resist being used as attributives. As your examples show, if you want to use an adverbial in this manner, you're required to mark it with the possessive -'s.