Про короля: The king is the source of all legitimacy. In real feudalism, no lord holds his land by right. He holds it by grant. The king granted the land to the duke. The duke granted a portion to the count. The count granted a portion to the baron. The baron granted a portion to the knight. The knight granted a portion to the serf. Every link in the chain derives its authority from the link above it, and the chain begins with the king. The king is the fons honorum — the fountain of honor, the source of all titles, all privileges, all jurisdictions. Without the king, the chain has no beginning. The lord's authority has no source. It simply is. And authority that simply is, without a source, is not legitimate. It is merely powerful.
The king is the ultimate judge. In real feudalism, the vassal who is wronged by his lord can appeal to the king. The king's court is the court of last resort. The king can hear the complaint, examine the evidence, and render judgment. The king can strip a lord of his titles, confiscate his lands, imprison him, execute him. The king is the one who stands above the feudal hierarchy and can correct it. Without the king, the vassal has no recourse. The lord is the ultimate authority within his own domain. There is no higher court. There is no appeal. There is no justice beyond the lord's own sense of fairness, which may be considerable or may be nonexistent, and which is, in any case, unchallengeable.
The king is the mechanism of reform. In real feudalism, the king can change the rules. He can grant new charters. He can revoke old privileges. He can create new lordships and dissolve old ones. He can reform the legal system, override unjust customs, abolish oppressive obligations. The king is the one who can look at the system and say: "This is no longer just. I will change it." Without the king, the system is frozen. The blood oaths are eternal. The Ancient and Noble Houses are permanent. The hierarchy is immutable. There is no one who can say: "This system is unjust, and I will reform it." Because there is no one above the system who has the authority to reform it.
The king is the unity of the realm. In real feudalism, the king represents the idea that all these competing lords, all these overlapping jurisdictions, all these tangled obligations, all these private wars and local customs and conflicting loyalties, are part of one polity. The king is the symbol and the substance of national unity. Without the king, the realm is not a realm. It is a patchwork of competing fiefdoms, each with its own lord, its own laws, its own obligations, its own private army. There is no "England." There is no "France." There is no "the Realm." There is only a collection of lordships, each sovereign within its own borders, each owing allegiance to no one, each pursuing its own interests, each warring with its neighbors.
The king is the check on the lords. In real feudalism, the king is the one who can punish a lord who abuses his power. The king is the one who can say: "You have oppressed your vassals. You have broken the king's peace. You have waged private war without license. And I will strip you of your titles, confiscate your lands, and hang you from the battlements of your own castle." Without the king, the lord is unchecked. He is, within his own domain, absolute. He can do what he will to his vassals, his tenants, his servants, his house-elves. There is no higher authority to appeal to. There is no one who can say "stop." The lord is a petty despot, and his despotism is total because it is final.