Although modern grammar doesn't recognize the difference between gerunds and present participles, if we were to distinguish them, then in your case we would have a 'gerund'. The clearest way to see this is to note that you can replace the accusative him by the genitive his, and the genitive can only work with the 'gerund'.
Discussion
On the one hand, as tchrist and Edwin Ashworth have said in the comments, modern grammar does not distinguish between gerunds and present participles. For example, CGEL has whole sections whose entire purpose is the defense of the proposition that those two are not distinct (pp. 82-83 and 1220-1222).
On the other hand, CGEL also says this (1220-1221; the asterisk, '*', means that what follows is not acceptable English):
A difference in internal form: case of the subject NP (noun phrase)
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.
This difference, however, is obviously relevant only to those constructions where the non-finite clause can contain a subject: it cannot be used to justify a distinction between 'gerund' and 'present participle' in the numerous constructions where no subject is permitted. In terms of our analysis, the contrast in the case of the subject is handled by our distinction between complement and non-complement gerund-participials: genitive case is restricted to the former, nominative to the latter. If the traditional distinction of gerund' vs 'present participle' is to be maintained, it must be based primarily on properties of the subjectless construction. But here there is no difference at all in the internal form of the constructions.
So: if the subject can be in the genitive, then what we have is a gerund. Now consider your sentence:
[1] We’re depending on his/him/*he finishing the job by Friday.
This is just like CGEL's [39i], and thus finishing is a 'gerund'.
Which form would native speakers most likely say when they watch football (soccer or American) matches from the comfort of their home?
In AmE, IMHO, your samples 1 and 2 are appropriate.
Is there a difference in meaning or usage if the speaker is thinking of their physical presence at the sporting event?
In lieu of your samples, I suggest:
I like going to football games.
I am not sure of semantic reasonings. In summary, "I like going to a football game" or "I like to go to a football game" (or match)? are one in the same. I wouldn't normally say "watch football in a stadium."
Hey, are you watching Monday-nite football tonight? I sure am! I am actually going to the game.
Best Answer
I don't know the right way of analysing this, but it seems to me to have to do with grammatical aspect. The Cambridge Grammar of the English Language (Huddleston and Pullum) interprets English as having two aspects, progressive and non-progressive. (The perfect is not considered to be an aspect; it's treated as a distinct system for various reasons.)
As far as I can tell, the gerund-participle is marked for progressive aspect, while the infinitive is not.
In your examples, the gerund-participle seems to agree in aspect with the form of the verb "do." (I don't know if "agreement" is technically the right term for this.)
I don't know why it has to match like this.
I found a resource about cleft sentences that just states this fact, but doesn't explain it:
–"Cleft sentences"; handout from the website of Università degli Studi di Padova, Dipartimento Dei Beni Culturali
According to this resource, this construction with "what" is often called a "pseudo-cleft" sentence to distinguish it from sentences using the dummy pronoun "it." They may also be called wh-clefts, apparently.
I found another resource, The Pseudo-Cleft Construction in English by F. R. Higgins, which says:
Unfortunately, I was unable to locate any other mentions of the progressive in this book (I initially had high hopes that it might explain how the construction interacts with the English aspectual system).
Some sources mention that some speakers do use the past participle form in sentences such as "What I had done was watch(ed) TV."
–"On the Syntactic and Semantic Properties of VP Foci in Pseudocleft Sentences in Japanese", Yuki Ishihara, p. 37
I tried to look up information about the internal grammatical structure of pseudo-cleft sentences, but everything I found looked very complicated.