How brutal was the daily life in the middle ages in terms of random violence?

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This may not be exactly what you're looking for, but it depends. A couple things before we dive in: the medieval period lasted a thousand years, and it can only be understood in terms of intense regionalism - beliefs, social and political structures, linguistic quirks, and common daily practices differed enormously between regions, and within those regions they differed enormously in different periods; second, The Last Kingdom takes place during a period of intense conflict, in which violence was common not as a result of any universal truth of medieval life, but because of the social and political conflicts engendered by the larger, overarching warfare. Any conclusions we might draw from the fictional situations of the characters must be understood first as the fictional portrayal of this culture and second as a portrayal of an unusually intense period of warfare and conflict. To use a more modern example, someone unfamiliar with the United States post World War II might watch The Godfather and ask "why was the United States following World War II so violent?" if they didn't understand the very specific context of the violence depicted in that film.

If there is a general trend in random violence in western Europe during the (vast and complicated) medieval period, it was that social structures discouraged, controlled, and punished it, but it should also be understood that the way medieval people tended to view and understand violence is somewhat different than today. Laws were often what we consider common or customary, meaning (being very general here) that the context of an action tended to matter much more than the fact of the action. Killing someone, for instance, could be justified in numerous ways, from self-defense to arguing that there was a provocation beyond the reasonable ability to withstand, to accident, and, as Trevor Dean's Crime in Medieval Europe argues persuasively, the in vs out-group dynamics of the action. In other words, someone "not from round here" is much more likely to be punished for a crime than someone local. It's important to point out the hyper-regionality of much of this. So much depended on the legal structure of the place concerned, how it was governed, the empowerment of its locals to take matters into their own hands, and the interest in the locals to punish whatever crime or act of violence may have occurred. There were also structural differences in how an accused criminal would be apprehended, housed, and tried, and much depended on that individual's standing in the legal, political, or religious structure of the area. Who you worked for, in other words, could make an extreme difference in how your action was viewed in legal terms. An appointed officer of a lord would be treated much differently than a common citizen of a town - which also assumes that there's something like citizenship in the region in question, which itself would not be universal and would matter a great deal - and the outcomes would also be similarly different.

Citizenship was a reciprocal relationship of privilege and responsibility. A citizen was likely to have access to legal, social, and financial structures that non-citizens didn't, which gave them some distinction and conferred a certain legal benefit of doubt if they were to get into trouble. But it also placed on them various burdens, from taxation and other monetary burdens, to work requirements on occasion (such as when a town or city built walls), to membership in the city guard or militia, which also meant keeping and bearing arms for the purpose. This meant that a town's citizens were the town peacekeepers and first responders. If someone needed to be arrested or apprehended, they did it, usually in cooperation with an appointed or elected officer like a constable or reeve of some kind. If not in person, citizens were sometimes required to contribute to a fund to hire men to go out and do the violent parts.

Much also depended on the political circumstances of the region at the time. Somewhere near a war was much more likely to see acts of violence than peaceful places. In the longstanding conflict of the Angevin and Capetian dynasties in Normandy, raiding and warfare would have been astonishingly common for whole generations, but further into the interior of both kingdoms it would have been much less so. Border regions were understood as more violent and chaotic than central regions and led to some counties (in a generic sense, not necessarily a piece of land ruled by a count) were "marches," or borderlands, and in some systems of lordship and aristocracy, the lords who were meant to rule those marches were given special titles and special privileges. The German term Margrave is in reference to a Marcher Count, and there were complicated social and political extrapolations for people living in border regions, especially in times of conflict, and then again, even decades or centuries after the borders of a nation, kingdom, or empire pushed far past those marches, the hereditary titles and nomenclature tended to remain, further complicating things.

To get a little more specific and to address one of your subquestions, would people commonly kill or cripple each other over minor inconveniences? Probably not. People in general everywhere and everywhen tend to prefer peace to violent conflict, but their version of peace might look a little different to ours. For one thing, practices that a lot of modern folks tend to assume are about no-holds-barred murderous violence - duels and feuds - were practices that were often attempts to control and restrain violence. Duels were sometimes ways to immediately nip in the bud any conflict that might threaten to become more widespread, and as such it tended to limit violence to a short period and to only the principals, but it also encouraged men to weaponize language in ways that encouraged more duels than just the ones that were important to keep peer groups together. There are far too many variations on duel subcultures to generalize here, but duels were about status and integrity and maintaining both far more than they were about killing your opponent. Fairness in duels was essential, because if someone dies in a duel that was determined to be unfair, it could create a bigger conflict as a result, which is one of the things duels are intended to prevent. And so scrupulous adherence to the unwritten rules that determined the fairness of an interpersonal conflict were paramount.

A similar dynamic occurs in feuds, which were, similarly, conflicts between two peers of high status - an armed peerage - in which the principals were extremely careful not only to articulate the basis of their conflict through the issue of a written statement - a "cartel of defiance" - prior to any hostilities, and then a supervised reconciliation after both principals were convinced to negotiate. What happened in between the cartel and the negotiation looked an awful lot like warfare. Kidnapping, arson, property destruction, and theft were all fair game, even if killing was relatively rare. Still, to a peasant caught in the middle, does it matter if your wagon was captured by bandits or by a robber knight in a feud? Probably not. The robber knight Götz von Berlichingen jokes in his autobiography that he had captured one hapless peasant on three different occasions. In another instance, he jokes:

The others were vain bale binders from Nuremberg, and I pretended that I wanted to lop off all their heads and hands. But it was not in earnest. Still I made them kneel down and lay their hands on a log, and I went down the line and there I would kick one in the ass, and another I would give a blow behind the ear. That was my penalty to them, and I let them leave.

Which is a neat way of demonstrating the utter hopelessness of men captured in the course of one of these feuds, even if, for the most part, peasants and merchants would have been left alone but for any ransoms they might be required to pay. And while Götz may have had a somewhat insensitive sense of humor, other men of his class were known or suspected to have committed atrocities against innocent men. Götz's friend and peer Conz Schott, for instance, allegedly murdered some officers of Nuremberg sent to him, and sent others home with their right hands severed. Perhaps because of the reputation some knights of his class had earned, Götz is extremely careful to frame and explain all of his feuds in scrupulous detail to justify all of his actions as legitimate. Nuremberg, for its part, was equally zealous in justifying its reaction to banditry, and executions of criminals were prominent displays of the city's power.