Whenever I read anything that purports to explain the social norms of the Victorian era vis-à-vis marriage, exactly how it was that couples got engaged and married is always glossed over. I’m curious to know both what would have been going through the minds of a new debutante, a single nobleman, their families, etc. as well as what historians with the benefit of hindsight think was actually governing their actions (as what people say they care about usually differs from what actually motivates them).
Were people just trying to find the richest or most titled spouse they could? How did an individual person’s physical attractiveness, personality, or “morality” figure into this? Was there room for someone poorer (but still in the gentry/peerage) to “marry up” if they were very charming or desirable? Or since everyone was apparently having affairs were these marriages purely for the sake of social status and furthering one’s lineage?
Could a man who didn’t find a wife in London one Season try his luck in Paris a few months later, or vice-versa? Would English and French nobility have cared about different things in a match? Was there a stigma attached to marrying a foreign (but still wealthy/noble) spouse beyond the widely known phenomenon of American industrialists marrying their daughters into the British peerage?
I’m still very interested in other time periods before/after the above, so please don’t interpret any of my wording narrowly.
Could a man who didn’t find a wife in London one Season try his luck in Paris a few months later, or vice-versa?
To start at the beginning for those reading in, "the Season" had a specific meaning to English society: the time when Parliament was in session, which was when the aristocrats that made up the House of Lords and the second sons and landed gentry who sat in the House of Commons came to town with their families - December to May. After this, they would go back to their country seats.
New York's social season was a similar period, although there was no clear reasoning along the same lines - most likely, it was from a combination of imitation and the fact that everyone wanted to be out of the city during the miserable summer. (The summer social season was in Saratoga or Newport, depending on the period.) Paris's social season was also in the spring, so you couldn't leave for France after London's was done and expect to find a similar round of events.
It's important to bear in mind that pretty much everyone who participated in the Season expected to continue doing it perpetually, and was able to meet individuals to marry at other times - visits to friends' country houses, while staying near other wealthy English people in the Italian countryside, etc. The trope of a protagonist (usually a heroine) whose family only has enough money for them to have one Season to get married, or who desperately wants "a Season", is a fictional construct intended to give some urgency to the plot. (Even referring to it as "the Season" doesn't seem to be as extremely common in nineteenth century texts as it is in, say, Georgette Heyer. It was just the time that was right for going to London.)
It's also important to bear in mind that attitudes toward the Season couldn't really vary by region or social status, because only a fairly small slice of the population cared about it at all. If you didn't have a male family member in Parliament or weren't so wealthy that you could afford to come to London for months anyway, it did not affect you. In the countless small cities, towns, and villages dotting the country, this time period didn't mean anything special - the social round went on throughout the year.
Were people just trying to find the richest or most titled spouse they could? How did an individual person’s physical attractiveness, personality, or “morality” figure into this? Was there room for someone poorer (but still in the gentry/peerage) to “marry up” if they were very charming or desirable? Or since everyone was apparently having affairs were these marriages purely for the sake of social status and furthering one’s lineage?
By the nineteenth century, it was expected that everyone was marrying for love, or at least for earnest mutual respect. Rank was not supposed to come into it, except inasmuch as people would only be likely to meet if they were of roughly the same social status. Parents were to offer advice and give or deny permission appropriately, but not to choose their children's spouses themselves.
Of course, that doesn't mean that's what always happened. Parental pressure certainly was a thing, and mercenary marriages certainly were a thing.
That being said, everything you've mentioned could easily come into it, but there's no way to generalize about the rationale behind different matches without just saying that all of these factors could come into play. I can say that the marriage market was not solely about trying to land "the biggest fish" or always to move up - that's a logical impossibility, anyway, since for one person to marry up, the other must be marrying down - but there was actually a good deal of fluidity in the otherwise rigid social system when it came to marriage. If you look through Burke's Peerage, you can see that people often married those a rung or two above or below, creating and interconnected network of the peerage and gentry. For instance, in the 1848 edition's entry for Viscount Midleton, we can see that the third viscount's children married the daughter of an earl, the daughter of a bishop, and the daughters of squires (a kind of descriptive/courtesy title for the untitled landed gentry - Mr. Bennet of Pride and Prejudice is a squire) - a huge range, representing a step up for the eldest son/fourth viscount, an equivalence for the second son/archbishop, and steps down for the younger sons with no titles. The fourth viscount's eldest daughter only married a country squire, even though her father had been given the title of baron by that point and was going to be the heir to the viscountcy; his second wife was the daughter of a squire herself, and one of their daughters married a clergyman. You see this in all of the entries in the book: there were vastly more members of the gentry than the lower peerage, and vastly more members of the lower peerage than the upper peerage, so it was inevitable that daughters of dukes would marry down and daughters of squires would marry up. It could very well reflect on their individual physical attractions, moral character, or dowry size, but ... we really don't know, because almost none of these matches left information behind on their courtships (or at least most letters or diaries they may have left have not been analyzed and published). It would take a significant amount of research to find out whether these are cases of dowryless Elizabeth Bennets marrying vastly outside of their circles or near-noble Georgiana Darcies with family ties higher up the social ladder simply stepping back up to their mothers' ranks.
Would English and French nobility have cared about different things in a match? Was there a stigma attached to marrying a foreign (but still wealthy/noble) spouse beyond the widely known phenomenon of American industrialists marrying their daughters into the British peerage?
I don't actually know because it's so rare to see international marriages outside of the Buccaneer mold you mention, which I suppose answers the question. Women would move to their husbands' locality, so a French woman who married an English lord would be expected to leave her family behind and see them rarely, which would be an unattractive prospect in most situations. In English/American fiction of the time, there was still plenty of xenophobic baggage for mixed marriages, but there was also often some class differences involved in those, so it's harder to judge.