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.
My opinion: plural except in a special case (see below). The only explicit statements I've found to corroborate my opinion are on Answers.com regarding subject/verb agreement and a chat board for college students, neither of which strikes me as particularly authoritative. Nothing I can find indicates that anything other than a plural is appropriate when the subject of the sentence is two of anything conjoined by "and," including two gerunds.
Special case: gerunds that go together to form a unit of activity: drinking and driving, or texting and driving, etc. In those cases, when the point is the combined act, then a singular is nearly always used. Now that I think about it, the singular or plural helps differentiate: "walking and chewing gum is a skill mastered by most people" versus "walking and chewing gum are physically active tasks, thinking is not, but all three burn calories."
Best Answer
What you're confronting here is one of those fiendish composite verbal constructions English employs with relish to keep schoolchildren, their teachers, grammarians of every stripe, and speakers of non-English languages of the Wrong Sort† in their respective places.
The construction here is [look forward to]. This must be carefully distinguished from the collocation [look (vb) + forward (adv.) + to (inf. marker)], which is how your first sentence must be construed:
Look forward to, employed as a fixed phrase, is a “phrasal verb”. It is transitive, taking a substantive — a noun, a pronoun, or a noun phrase — as its direct object. It may be glossed as “happily await” or “expect” or “anticipate”:
There are two forms of a verb —celebrate, in your examples— which may act as a noun, and thus as the direct object of a verb such “to look forward to”:
But even English has to draw the line somewhere — “look forward to to” is just too awful (in any sense) to contemplate. Consequently, look forward to takes the gerund, as in your second sentence:
There's far more to parsing this insidious phrase; but you need not yet be troubled with the descriptive grammarians' cunning distinctions between prepositional and particular phrasal verbs. And you certainly don't want to be involved yet with the difficulties of transforming this sentence, into which you may be drawn if your editor is of an Enlightenment persuasion and deprecates ending clauses on a preposition because that's not done (nefas) in other languages of the Right Sort†. It takes an iron determination (and a strong stomach) to come up with constructions like this:
In any case, the celebrate in your first sentence is not Future Simple, and the celebrating in your second sentence is not Future Continuous; they are, respectively, an infinitive and a gerund.
† The Right Sort of non-English language embraces the Greek of Pericles, the Latin of Cicero, and possibly the Hebrew of Moses, who may be regarded as a British Israelite denied the opportunity to realize his Britannicism (just as was he prohibited from entering the Promised Land — in fact, the two may be the Same Thing). All other non-English languages are of the Wrong Sort.