Unless the person is of the very lowest class, i.e. a serf or totally landless peasant, then yes, they probably would have some kind of access to a horse. As you note, we associate owning a horse with being rich, but in the premodern world, they were obviously far more important than they are now and most people would have either owned a horse or been able to hire one if they needed to go on a journey. There are different types of medieval horse, such as the destrier or charger (the very expensive warhorse that was only owned by upper-class knights and nobles), the palfrey (general riding horse, middle-upper class) and the rouncey (general riding horse, middle-lower classes). You also had packhorses, workhorses, mules, etc. If you were traveling any distance, you would be able to hire horses; otherwise, if you were a member of a lord or knight's retinue, it was his job to provide you with one. It's kind of like owning a car today -- i.e. people of all economic classes have cars of different types and values, but they do usually have one, because it's just a general necessity for getting around.

However, I find that when most people ask about the medieval era for fictional/writing purposes, they're generally picturing something roughly similar to the thirteenth/fourteenth century, and not the earlier medieval era. An early tenth-century thegn of mixed Norse/Anglo-Saxon descent definitely doesn't think of himself as an "Englishman," for one thing, but his descendants are probably doing so a few hundred years later (especially at the outset of the Hundred Years' War with France). But said Anglo-Saxon thegn still lives in a culture that is intensely horse-centric (think the Riders of Rohan) even if he doesn't have as many places to go with it, his world is generally much less centralized, he would have little reason to go to London anyway (at this time the capital of England was Winchester, the seat of power for the kingdom of Wessex), and so forth. In short, while the organization of his world and its social, political, and cultural expectations are very different, he probably still does have access to a horse, as it's an important part of his identity. So this is a good example of how while the medieval world does change drastically over half a millennium, like any human society, it still maintains certain core communalities. There was never a time when horses were not important or inaccessible (see above: they're like cars) even if that changed with the times.

pros: yay! i finally look like a guy! people think im a man!

— Who are you, Louis? Who are you outside of me? If there was no me, and there was no him, who would you be? What do you want? How you gon' get there? (Interview with the Vampire, Season 2 Episode 2)

This is really good, and I think there are a number of narrative facets in play that help explain it, the first one being the relatively straightforward fact that sexual violence is at the heart of Muir's work and has been all along; as the series begins to seriously mobilise its themes in coming towards its close, it makes sense that explicit references to sexual violence would move to the fore of the narrative to increase the pressure around that thematic nodal point. Nona, to me, felt like a book that was rife with references to sexual violence not only in the forms that you signify but also in, for example:

The language used to describe how Nona relates to her body and her selfhood; the image of her with her hand clutched between her thighs wishing that she could mutilate her body enough that nobody except herself could ever 'want' it and force her to rescind ownership over it, then developing into the entrapment that Alecto feels in a body that she was made to live in against her will;

The connection drawn between John's resurrection(s) + reanimations and sexual violence, figuring each act as boiling down to the same essential discourse; the language used to describe the reanimation of U— and T— (Ulysses and Titania) includes references to them as '[his] kids,' and the image of their being 'breached' with a thermometer causes the process to double as one of penetration; the 'creation' of Alecto reads as something close to a rape (arguably a play on the double meaning of 'rape' as pillage and 'rape' as sexual assault, a very easy in for interrogating the relationship between colonial and sexual violences), eg. 'I was terrified you'd find some way to escape before I was done,' and of course 'Fuck, Marry, Kill' becoming 'Marry, Kill, Reanimate.' I explained all this in more depth in the linked essay, hence why I've just glossed it here lol

That the entire John section is, to me, one long reference to Lolita right down to the unreliable narration, the standouts being: that significant conversations take place on a beach (the 'kingdom/princedom by the sea'), the 'creation' of Alecto that mirrors the constructing of Lolita, the J+E/A/H again echoing how names as signifiers of sexual ownership are used in the book;

That, corroborated with the reading of John and Alecto's relationship as having been (definitely metaphorically, perhaps literally) one of sexual abuse, the thing that, to put it crassly, 'makes' Alecto and 'unmakes' Nona – the thing which distinguishes the two – is the reminder that 'John loves [her], John needs [her].' (Crucially, it's not just the naming here – by the time this line is spoken, Kiriona has already referred to Alecto as Alecto out loud.) Assuming that this plays with the (accidentally?) sinister line in Annabel Lee which runs: 'And this maiden, she lived with no other thought / Than to love and be loved by me,' we see the last third or so of Nona run up to Nona remembering something that terrifies her to confront: "I’ll be different. I’ll remember everything … I’ll remember the thing I’m trying to forget. And Palamedes—I won’t love him. I won’t love Camilla, or Pyrrha, or Hot Sauce, or even Noodle. I won’t love anything … I won’t know how. I won’t be me at all, or … I’ll be the me who knows the thing." I can't find the post so apologies for lifting someone else's take, but a while back someone else noted that Nona knows how to kiss gently – as she does to Kiriona – whereas Alecto only knows mouth-biting and 'how meat loves meat.'

I could go on and on and on about how Nona is a book about sexual trauma and the conditions under which sexual violence is made both possible and socially passable, but like, I've done that elsewhere. The point is: Nona is a book about sexual violence, but it's able to be a book about sexual violence because the groundwork was already laid in its predecessors. The immediate textual presence of sexual violence in Nona feels a lot less jarring or abrupt when you consider, eg.:

The figuring of the necromancer/cavalier relationship as one necessarily of inequality, discursively built from the relationship between John and Alecto (which is in turn built off of Lolita's rendition of Annabel Lee…..); that cavalierhood as a subject position, when done 'correctly,' invites a certain measure of sexual objectification (cf. Cytherea) as part of that degrading practice;

Gideon and Cytherea, specifically, is the twofold grooming literal (through the seduction of an eighteen year old on the part of someone significantly older than her under false pretences figured with predatory and fetishistic language) and grooming into 'correct' cavalierhood (with 'correct' cavalierhood in turn then leading to, well, death);

John and Alecto as Annabel Lee and the unnamed narrator, which in turn figures them as Humbert and Dolores; similarly, John's positioning himself as a surrogate father to Harrow in Harrow (and how closely that practice echoes The Magician's Apprentice!).

(Sorry for all the bullet points, it's just the easiest way to organise my thoughts without dropping more big blocks of text on the dash than I have to.) Like, it doesn't feel anywhere near as out of left field when you notice that the thing the text has been building from all along is a social paradigm to which sexual violence is an essential property.

And like, besides a general gesture towards 'developing themes,' I think the bluntness with which sexual violence is placed in front of us in Nona comes in tandem with the fact that Nona is the book where the voice and action alike as we receive them are no longer limited to the solipsism of empire. Nona is the first time we as readers are permitted to properly move out of the imperial core, and in giving us a glimpse of an occupied planet (and references to other such occupations), we get as straightforward an account of what living under imperialism looks like. Like, Nona succeeds in giving its cast of non-empire characters rich and wilful and agentive lives without shying away from the facts of their situation; if anything, these two aspects are woven together to great effect. Before now, we've only ever received accounts of the empire beyond the core, ie. occupied land, through imperial figures, and were expected to deduce that the occupations taking place were, as would be any imperial occupation, violent; Nona removes a cultural shield that existed in the first two books, and makes that violence about as explicit as it can possibly be.

In the empire itself, there's a somewhat fantastical sheen to the world the books occupy that makes it almost a little ridiculous at points. It's highly aestheticised and atomised (down to the Nine Houses with their corresponding cultures and colours and elaborate titles and numerical surnames), and that aestheticisation is archaic: swords, chivalry, duchesses and ladies and lords and the language of a feudal nobility to which fascist aesthetics can often turn. Sexual violence takes place through elaborate socially sanctioned relations within the nobility (Gideon and Cytherea and John and Alecto – and, I would argue, Corona and Ianthe – each as paradigmatically necromancer/cavalier) such that the fact of it can be kind of … kicked under the carpet, at least diegetically. I think there's real weight thrown behind what John says to Harrow in one of the John chapters:

He said, I guess you could say … we had beef.

When she did not laugh he said, “I can’t believe nobody’s ever going to laugh at my jokes again. I can’t believe it. It’s all gone, I’m the only one left. It’s just me and you and no more jokes.”

Like … it's stopped being funny. The jokes don't land anymore; John's "She didn’t laugh. I laughed. He said, I guess I’ve always thought any pun was automatically funny" echoes Gideon's having "assumed that puns were funny automatically" from two books ago (lmao, like father like daughter!), but the key thing is that this time the joke doesn't land the way the one about Gideon was able to. It's an immensely revealing echo of what was originally just a slightly cringe throwaway line – the narrative terms have changed. In-universe, this is because we're meeting John in what is effectively a wasteland after the death of his last three friends; for us, this is the point where a lot of the kind of silly stuff from the last two books (the jokes, the memes, the skeletons, whatever) starts hitting a brick wall because those things were intended as obfuscatory devices occluding violence. (Nona is definitely still a funny book, but it lacks that particular, like … campy silliness that the other two had. Which, imo, is to this exact purpose.) My point is, like, Nona is blunt about sexual violence because Nona is the point where obfuscation and deflection and cultural normativity all just stop working.

I once saw @olreid refer to Alecto as 'the madwoman in the attic of the Nine Houses' and that phrase has stuck with me for MONTHS, because – I mean! Nona is about finally confronting the things that John's elaborate cultural constructions and elevated language were trying to shunt out of sight. Sexual violence can hover at the edges of the narrative in Gideon and Harrow, appearing to us through elaborate cultural rituals such as those taking place between Gideon and Cytherea or through John's references to Poe's poetry, in a manner that diegetically disguises the violence as violence, but Nona is Muir saying, like – you got what this was, right? You got what was really going on here? If I take away the cultural gloss placed over it, will you see it for what it is?

So like, in short: sexual violence emerges abruptly and explicitly in Nona because it is the point at which Muir begins to really heavily foreground a theme that she has been sowing throughout, and a lot of the barefacedness of the fact is in keeping with the barefacedness of violence in the broader sense throughout the book, as we move away from the solipsism of empire and are finally asked as readers to grapple with its material consequences.