Do we know any Mayan history directly from their own writings? Do Mayan writings give us names and dates of historical events?

Yes - almost everything we know about the details of Classic Maya history (ca. A.D. 250 - 1000, although writing dropped off pretty precipitously after A.D. 830 or so) comes from their inscriptions. Those inscriptions do give the names of people (almost always kings) who did things, typically pretty laconic descriptions of what they did, and the dates, specific to the day, on which they did them. However, Classic Maya historical writing is often frustratingly different from the kinds of history you might be used to reading. Furthermore, ancient Maya authors wrote about the past in different ways depending on what kinds of events they were discussing and how long ago those events occurred or were believed to have occurred. There is a chapter about this in a recently published book, Thinking, Recording, and Writing History in the Ancient World, edited by Kurt Raaflaub, and I'd encourage you to check out the whole book. The chapter ("Sources and Scales of Classic Maya History") is my source for the following summary of what ancient Maya history is like.

Most surviving inscriptions record events that had just happened, or the act of inscribing was itself the event being recorded. These kinds of texts include the so-called "Primary Standard Sequence," a group of conventional phrases that were inscribed on fancy ceramic drinking vessels, which say things like "Thus is dedicated the painted surface of the drinking cup of so-and-so." Most stone monuments were commissioned for important dates in the Long Count calendar, equivalent to the ends of decades or centuries in our calendar. Their inscriptions usually record that a king celebrated the occasion by putting up the monument. Sometimes they also provide background information, like who the king's parents were, or how his ritual acts followed precedents set by earlier kings.
Some other texts, mainly in the western part of the Classic Maya script area (the area around the Usumacinta River), give accounts of people's lives and careers. For example, a few texts from the site of Piedras Negras provide a brief biography of a queen named Winikha'b Ajaw, focusing on her "presentation" to the king (then the crown prince) and the birth of her daughter. Or then there's the so-called "Tablet of the Slaves" from Palenque, which deals with the military victories of a local lord (not a king) named Chak Suutz'. These kinds of texts were written with some practical, political purposes in mind, which guided the subject matter. For example, Winikha'b Ajaw's official biography seems to set up her daughter, Juuntahn Ahk, as the heir to the throne, or, more likely, as the woman that the man who next became king would need to marry.
Again in the west, most monuments that aren't primarily about calendrical rituals deal with military successes, especially the capture and presentation of prisoners of war. Prominent captives are often named in these texts, which could accompany scenes showing the moment of submission or defeat. After capturing some notable person, kings could call themselves "the guardian of [captive's name]" in other texts. At Yaxchilan, martial inscriptions focus on these kinds of capture events, but in quite a few cases throughout the Maya lowlands there are somewhat longer accounts of military campaigns. These are not very detailed, but they do tell us things like "the king went out from Town A" (presumably under duress) and then "went up to Town B," or "the king settled at Place C" and then the next day attacked Town D. Subordinate commanders are rarely mentioned, and to whatever extent commoners were part of Maya military forces, they are totally left out of the inscriptions with the possible exception of a lovely expression for the mass murder of prisoners: "heads were piled up into a hill, blood was pooled."
When texts look back over multiple generations, there's some kind of a more or less overt political point to the story. One good example is an altar, now in the Dallas Museum of Art, which tells the tale of how the low-ranking but faithful lords of a site called Sak Nikte' married princesses of the powerful and prestigious Kan dynasty over a couple of centuries. To me, the most interesting example is a carved throne back from Piedras Negras, which depicts a king of Yaxchilan and his son attending a party held by a Piedras Negras king. By way of background, the Piedras Negras and Yaxchilan kingdoms were normally bitter rivals who fought to control the Usumacinta River, an important transportation corridor. Anyway, the king of Piedras Negras shown in the scene is probably the grandfather of the king who commissioned the piece, so the scene is itself historical in that sense. But what makes it especially neat is that this earlier king is shown giving a lecture to the king of Yaxchilan about even earlier events, basically saying, "Remember, your grandfather was a vassal of Piedras Negras, and ruled by the consent of my predecessor." Even better, this is the only source of information about that king of Yaxchilan: he is never mentioned on any surviving texts at Yaxchilan himself, and in fact the odds are that a later king did his best to wipe out all record of the guy.
At a much larger time scale, Classic Maya historical writing styles change again. When they wrote about things people were supposed to have done in the Preclassic period (before A.D. 250 or so), Classic scribes had only a few kinds of acts in mind. Subjects of these kinds of histories are always gods or kings, sometimes both, and they don't go to war with each other - there's no question of political opposition to their authority. Instead, they do things that establish precedents later kings have to emulate, like adopting new names upon taking office, or founding dynasties, or slaying cosmic monsters who gush oceans of blood. I guess that last would have been a tough act to follow.

No matter what the scale of time involved, Classic texts are always very precise about the day on which a thing happened. That's in large part because the Maya had a 260-day divinatory calendar, with each day having a unique name. It was felt (1) that days in that calendar determined the kinds of things likely to happen on them, whether good or bad, and (2) that days with the same name were in some religious sense the same day. You'll never see a Maya text say something so vague as "In the beginning..." or "A long time ago...". Another way in which Classic Maya history is weird, from a modern perspective, is that it contains no explicit statements about why people did what they did. Motivations are sometimes strongly implied, as in "revenge narratives" about wars between kingdoms, but even then, it's just "On this day, those guys attacked us. Three score and twelve days later, we burned their city. This king oversaw it." Also, in violation of subreddit rules, ancient Maya texts never cite sources of historical information, whether existing texts or eyewitness accounts.