In ‘the musketeers’ by dumas, duels were fought in 1600s France over things as small as picking up a handkerchief or insulting a horse. How accurate is this?

/u/Georgy_K_Zhukov's linked answers are always worth reading, but I want to nitpick just a bit about what the duels, especially those in the early portion of the book, were about. Because while you're not wrong that the initial interaction of both of these examples involved a horse and a handkerchief, the actual "fighting words" were a result of escalation to unforgivable insults, which followed a fairly common pattern. Let's go through each example one at a time.

D'Artagnan's Horse of an unusual color

The horse, a Béarnese pony, was a gift from D'Artagnan's father. We are told, before the altercation at the Jolly Miller, that even though the horse was capable of bearing D'Artagnan the eight leagues a day he was hoping for, the unusual yellow-orange color of the fur was amusing to those he passed. D'Artagnan, of course, is a perceptive and quick-witted character, and so this bemused regard was not lost on him as he rode.

Once he arrived at the inn, he noticed a subdued exchange from men at a table near the inn's window:

But there, as he was alighting from his horse at the gate of the Jolly Miller, without anyone—host, waiter, or hostler—coming to hold his stirrup or take his horse, D‘Artagnan spied, through an open window on the ground floor, a gentleman, well-made and of good carriage, although of rather a stern countenance, talking with two persons who appeared to listen to him with respect. D’Artagnan fancied quite naturally, according to his custom, that he must be the object of their conversation, and listened. This time D’Artagnan was only in part mistaken; he himself was not in question, but his horse was. The gentleman appeared to be enumerating all his qualities to his auditors; and, as I have said, the auditors seeming to have great deference for the narrator, they every moment burst into fits of laughter. Now, as a half-smile was sufficient to awaken the irascibility of the young man, the effect produced upon him by this vociferous mirth may be easily imagined.

One of the elements that Dumas slides in to this novel is that each of the four main characters are avatars of a specific temperament. D'Artagnan is a textbook choleric: he's intelligent, a natural leader, full of energy and creativity, but he is also hot-tempered and extremely quick to anger. Choler being the humor associated with yellow bile, even the color of the horse is a joke about D'Artagnan's innate character. So when he sees someone making fun of him, his instant, overriding response is to confront them. He shouts through the window, demanding that the man respond, and he's ignored. The man continues making fun of him, which only makes D'Artagnan even more angry.

the gentleman made one of his most knowing and profound remarks respecting the Béarnese pony, his two auditors laughed even louder than before, and he himself, though contrary to his custom, allowed a pale smile (if I may be allowed to use such an expression) to stray over his countenance. This time there could be no doubt; D’Artagnan was really insulted.

When confronted, the man ignored him, and then doubled-down on the insult to his chucklehead friends. It confirmed, to D'Artagnan, that the man was deliberately snubbing him. Until that moment, it could easily have been a mistake, or could easily have led to the mysterious stranger simply apologizing, or finding some neutral way of claiming that he didn't intend insult to the young Gascon. This was a known language between men, and while it wasn't free from ambiguity, it did acknowledge the possibility of mistaken impressions, and allow two men to clear up possible confrontations without either man losing face.

Still, D'Artagnan wasn't beyond reconciliation. If the man simply said something along the lines of "I did not intend to insult you" and then offered to share a drink, even a hotheaded Gascon would have a hard time finding grounds for a fight. However, his adversary refused to back down, and instead tells D'Artagnan: "I was not speaking to you, sir." Dumas tags this exchange with the social meaning: D'Artagnan was exasperated by "this mixture of insolence and good manners, of politeness and scorn." In other words, the guy in the inn was being a dick on purpose. When he came out, his placid calm demeanor against D'Artagnan's buzzing energy made the confrontation even more funny to the audience.

And here's it's important to make a point that many of these duels and confrontations were driven by an audience. It's one thing to share a private moment with a peer or even a rival, but when you're being watched by other men who speak the language of honor, things become more important. D'Artagnan sense this, and at this most recent round of chuckling, he pulls his sword out. Unfazed, his rival explains that the horse has a funny color (usual among flowers but rare among horses), and that he does not often laugh, but retains the right to do so when he pleases. D'Artagnan tells him that "I will allow no man to laugh when it displeases me!" having previously tied the insults to his horse to insults to his person. His rival says, in short, "you're right!" and then turns around to leave.

And this is the breaking point. D'Artagnan's insulted, allowing the guy a few opportunities to clear things up, and each time, the man refused to do so. He doubled-down on the insult by treating D'Artagnan as, essentially, a boy. He is not worth apologizing to, is not serious enough a threat to treat as a peer. By turning his back on him, he was showing contempt, and to a fiery Gascon with something to prove, in front of an eager audience, it's too much of an insult to bear, and he threatens to attack the man from behind. When he turns, D'Artagnan lunges at him. The response is interesting, because D'Artagnan's rival - whom we later find is the Comte de Rochefort, an adversary who appears in the novel several times - actually seems to finally treat him as an equal. He leaps back, and then draws his sword and salutes. But then, his goons, the chuckleheads from the tavern, come out and beat D'Artagnan down with sticks. This of course didn't clear the stain upon D'Artagnan's honor, and it's one reason why later on in the book, D'Artagnan flings himself out of windows to get at Rochefort.

So in this sense, while it's the horse that gives the initial confrontation, it's Rochefort's refusal to acknowledge D'Artangnan's social station - a gentleman - that leads to what might have been a duel.

The Handkerchief of Aramis

This one is a little simpler, but still relies on a similar social dynamic: D'Artagnan, the junior, the newcomer, the fish-out-of-water craves respect as an equal but is too inexperienced not to make gaffes. Dumas tells us that when the Gascon approached the group in which Aramis was conversing, the group fell silent. D'Artagnan noticed that his appearance was an intrusion, and wished to make as little interference as possible, but couldn't just walk away, because that might seem insulting. Having previously seen Aramis drop a handkerchief, he tries to point out that Aramis had, certainly by mistake, trod upon a handkerchief had dropped as a way to accompany his withdrawal. He bends over to pick it up, pulls it out despite Aramis trying to stamp his weight down on it, and offers it back to Aramis.

But of course, he made a mistake. Aramis claimed that the handkerchief wasn't his, and it hadn't fallen from his pocket, and after extended skirmishes of politeness, in which Aramis continued to claim that he had not let the handkerchief fall, D'Artagnan uttered the fighting words:

“And thereby you have lied twice, monsieur, for I saw it fall.”

It's not the handkerchief. It's the lie. To be accused of lying was to be accused of being no gentleman, to say that one's word was untrustworthy. It was social death, and so Aramis had to respond, even if it was generally acknowledged that D'Artagnan was certainly no peer, he was at least a lower-sort of gentleman, and had already had an interview with the captain of musketeers. So Aramis couldn't simply walk away or ignore the insult, he had to respond.

All of this conforms, in a comical, exaggerated sense, to the culture of honor in early 17th century France. Dumas is obviously playful, and so there are specific comedic beats within these altercations, and he can't resist allowing his characters to fully embody their humoral temperament to the hilt, but, generally, their gentlemanly concerns are quite similar to the reality. Insults had to be addressed and confronted. To challenge a man's honor was a serious accusation, and almost always drew a response.