Getting right to the heart of the matter:

"What was chivalry, actually? How important was it?"

Chivalry was a set of norms, one might say an informal code of honour, which can be seen in various forms throughout the thought and writings of those who lived in Europe during the latter half of the middle ages (very broadly understood). Like all honour codes or sets of social norms, the extent to which it was followed or the importance given to it by individuals varied enormously.

To give an analogy to illustrate what I mean, we can think of 'chivalry' as what it means to be a good knight; it wasn't a written code or rulebook, but a code of values that would have been absorbed unconsciously since childhood, through stories, watching others, instruction by adults etc. Very much in the same way as modern Westerners know what makes a 'good cop', a 'good politician' or a 'good christian'. But of course, once we grow up, we realise the world isn't so black and white.

So, the TL;DR: to your questions is that how important chivalry was depended very much on the individual in question and we have evidence both of rather despicable and honourable actions by knights in the course of the middle ages.

I'm going to add, on a slightly tangential point, that chivalry was not set in stone, and that the parts which were emphasised in medieval culture changed over time. The most important point is that I don't think there was ever a time in medieval culture when a knight didn't want to fight (although I'm open to correction, especially from the vernacular literature, with which I'm not very familiar). Although a knight's participation in warfare was to take place under a strict moral code, fighting, warfare and violence were and integral part of his identity. The Latin word for knight throughout the medieval period was miles, literally meaning 'soldier' and the root of our modern English word 'military'. In many cases, such as in defense of his family, the church or in the service of his lord, the knight had a moral obligation to fight and not to do so would have caused great dishonour.

In the early days of 'knighthood', from the mid 10th century to about 1100, the emphasis in the emerging 'chivalric' ideal was very much on limiting violence and, increasingly, associating knighthood with Christianity and service to the church. The 'Peace of God' and 'Truce of God' movements, widespread in southern and central France at the start of the 11th century, used large public ceremonies to force knights to swear to limit violence to certain times of the year and never to attack widows, orphans, the poor or the clergy. This type of thinking lead into the development of the crusading ideology, which becomes explicit with Pope Urban II's sermon at Clermont in 1095, in which he urged knights to stop killing their fellow Christians and to wage war on the Muslims occupying Jerusalem instead. During the period of the First Crusade, knightly bona fides were established through violence, such as leading the charge against the Muslim enemy or, in one case, single-handedly charging a bear to save a defenseless peasant. That campaign, of course, also saw some horrific slaughters, most notably the sack of Jerusalem, where the streets of the city were made slick with blood.

You'll notice that this earlier 'chivalry' has little of what we think of about knighthood, with damsels in distress, courtly love and noble quests. This are tropes which developed from around 1150 onwards, particularly in emerging genres of vernacular epic literature which were performed out loud as entertainment. They constitute a huge topic of study, which I've only ever read the tip of, so I'm not going to comment much on how they ideas developed from then on. If someone who knows the literature better than I do wants to chip in though, I'd love to hear about it.

If you'd asked this question in a couple of months I'd have a better and more comprehensive answer, but what I'll do here is give a quick overview of what I currently know and suggest a couple of books that should help you understand chivalry better. I've yet to read the books myself, but both have an excellent reputation among academics, so that suggests they'll be both accurate and useful.

Our modern view of chivalry seems to largely be informed by post-medieval interpretations of medieval literature. I usually blame the Victorians for this, but I suspect that in this case the misconceptions predate them. From what I've read so far, chivalry was a malleable concept with, at first, far more practicality embedded in it than later depictions suggest.

For instance, John Gillingham points out in War and Chivalry in the History of William the Marshal that William's advice to Henry II that he disperse his troops, only to have them secretly regroup and launch a pillaging raid through the defenceless French territory, was considered "most courteous", and that the tactic of the chevauceé was normal for the time period.

Another incident in William the Marshal's life, where he takes £48 off a monk running away with a nobleman's daughter, shows that William's view of chivalry is very different to our modern conception of it. In the modern view, William would probably have escorted the two back to face justice and not merely stolen the money and let them go on their way. Some of his companions even thought he should have stolen the runaways' horses and luggage as well.

So, as you can see, chivalry of the period is clearly not what we would consider it to be, based on songs and stories from the medieval period and their later reinterpretations.

If you want to read more, then Maurice Keen's Chivalry and Matthew Strickland's War and Chivalry should provide you with plenty of context and information.

Edit: also, Practical Chivalry in the Twelfth Century: The Case of William Marshal