With examples from the Temeraire series and an unknown fandom (I don't know who Ross, Geoff, and Tom are, but they sound pretty hot).
The original question:
From a story I'm reading: "That was close," Ross said, and Geoff wasn't sure whether he was referring to Tom turning up or the interrupted fuck itself.
Should that technically be Tom's turning up? What is this usage called? Is there a more formal/less formal version of it, or is it always wrong not to use the possessive there? It seems uncommon to see it done that way in modern informal writing.
From Naomi Novik's Black Powder War: "I trust I need not insist upon your confining such sentiments to your breast." I'm sure that's correct, but most people would say "you confining" rather than "your", wouldn't they? (If we pretend they'd say something like that at all!) Another character on the same page says "Then they can't stop us going, either." Us not our?
I might say, "Everything went well except for him forgetting to do x": should that be "his forgetting"? It sounds stiff and awkward, but I'd at least like to know if I'm choosing to do it wrong!
Answer: Gerund is such a scary, scary word, but it's what this question is really all about.
clytemnestra215 said it best here when she explained:
THE STORY OF THE GERUND:
Once upon a time, there was Verb. Verb was very, very sad, for Verb didn’t want to show action; Verb wanted to be something! And so Verb wished upon the Star of Morphology, and a miracle occurred: Verb became an idea of action instead of being the action itself, and (as ideas are things) that meant Verb had become a noun! Oh Happy Day! And so Verb was given a new name: Verbal Noun. And Verbal Noun was happy forevermore, being able to control objects and doing other very fun things which nouns may do. The End.
Back to zebra363's question. The phrase referring to Tom turning up is, as you suspected, erroneous. "Tom's turning up" would be technically more correct. "Turning" is a gerund, or verbal noun, so "Tom turning" gives us two nouns in a row.
In the sentence I trust I need not insist upon your confining such sentiments..., "confining" is again clearly a gerund. It's functioning as a noun, specifically as the object of "upon." The sentence is thus correct. But notice how formal it sounds.
The next sentence (Then they can't stop us going) is an example of less-formal speech. Technically it's erroneous – to maintain the gerund status of going, it should say, "Then they can't stop our going." While you wouldn't want to make this mistake in narration, it would be perfectly natural in the speech of many characters, while the correct version would sound funny for some characters to use.
Finally, in your example, "Everything went well except for him forgetting to do x," you're absolutely right: "his forgetting" would be more correct.
In general, I think the reason people are prone to make this mistake, and the reason it doesn't usually set off alarm bells when we hear it, is that English gerunds look exactly like present participles (which are verbal adjectives).
"I’m puzzled by Temeraire's obsessing about Lawrence's well-being." [obsessing is a gerund (verbal noun), and "Temeraire's" is a possessive – Temeraire himself is pretty damn possessive, too, come to think of it]
"There's Temeraire, obsessing about Lawrence again." [obsessing is a participle (verbal adjective) modifying Temeraire, and Temeraire himself is a bit of a pervert]
If you've written a sentence for a character to say, and it contains a gerund in this kind of situation, you have some choices. If the character tends to speak formally, use "his." If the character would tend to speak informally, use "him." If neither one sounds right, rewrite the sentence: "Geoff wasn't sure whether he was referring to the fact that Tom had turned up or the interrupted fuck itself." "Then they can't stop us from going." "Everything went well, except that he had forgotten to do x."
[I'm dying to know what it means to do x. I'm imagining something really, really pervy.]