I'm trying to remember if Trip ever called Archer by his first name...not coming up with anything. I think he uses "Cap'n" as a term of endearment, in a way, though...even in the most intimate of circumstances, such as the scene where the two of them are talking together in "Observer Effect" and Archer is telling Trip not to give up.
The way I saw the show, T'Pol and Archer developed solid trust and friendship by mid-Season 2, but I wouldn't expect her to call him anything but "Captain," even in private. She's a Vulcan, and all the series have shown that typical Vulcans display a certain formality, especially with colleagues. I don't recall Tuvok ever calling Janeway "Kathryn" on Voyager, but it's been quite a while since I've caught an episode of that show.
In "Twilight" (which, as I understand, took place in an alternate future, not a hallucination), I got the impression that T'Pol called him "Jonathan" because she was no longer his science officer or captain, but his companion, caregiver, friend. There was no more professional distance to require such formality.
In TOS, Spock called Kirk "Jim" on occasion, but usually when he was under emotional duress. That famous scene in "Amok Time," when Spock discovers Kirk alive at the end, and he breaks into a grin--"Jim!"--then does his best to act as if it never happened, is priceless.
One of my favorite Trek novels from way back is "The Vulcan Academy Murders" by Jean Lorrah. The Big Three go for an extended off-duty visit to Vulcan, where the social custom practiced by Vulcans is to call each other by their given names. So Spock and Sarek, and even T'Pau, call Kirk and McCoy "Jim" and "Leonard" without batting an eye, something they would never do aboard Enterprise while in their professional capacities. Nice touch.