I apologize for your not receiving a response so far. But I think it is because you are asking very, very large questions that entire books could and indeed have been written about - the Inheritance of Rome or Framing the Early Middle Ages (both by Christopher Wickham) would be a great start.

Take, for instance:

In addition to this, why were medieval realms in Europe so decentralised with vassal lords all allowed to pursue their own interests instead of having one central bureaucracy managing most things, including the military, like in Ancient China?

Just to even answer this, a user would need to be intimately familiar with the whole of post-Roman European history, the rise of the successor states, the Merovingians and Carolingians, the development of the high medieval aristocracy, etc. Ideally, he would also know enough about ancient China to evaluate the comparison. There are debates inherent in these various periods which would require more than a brief skim to meet askhistorians standards.

I am prepared to speak a bit about military organization, recruitment, etc. It is true that medieval armies were ad hoc affairs, raised for a purpose and a limited time frame. This does not mean that soldiers were ill-trained or amateurish. Indeed, given the small size of medieval armies (10,000 is large; 20,000 huge; 30,000 immense), it would make little sense to bring wholly untrained men along. 20th century-style mass conscription was simply not a thing. With the exception of defensive emergencies, men who wanted to fight went to war. Now, this could be a full-time soldier (a knight, a lord's retainer, a mercenary), or an amateur with some military training from a friend or relative. What we know (and /u/mi13 can speak to this far better than I can) is that weapons were not rare, violence was not uncommon, and there was generally a more than adequate pool of trained military manpower to meet the demands of medieval war.

You may find this older post of mine enlightening. It's in reference to 11th-12th century Anglo-Norman military organization.

. . . I'd like to examine the issue in a more focused, detailed manner by concentrating on one place in a fairly limited time frame: Anglo-Norman armies of the late 11th-12th centuries.

I'd like to make a few cautionary notes first. One is that almost all of our sources are weighted very heavily towards the deeds of the great and the glorious, with much less emphasis on the routine of military organization and soldier life. Tangentially, this means that we know rather more about the (often) aristocratic cavalryman than we do about the foot soldier.

The tactical building block of the Anglo-Norman heavy cavalry was a unit called a conroi. Each conroi contained around 15-30 knights, though it could be considerably less. In theory the conroi was composed of men from the same geographic area, very possibly servants of the same lord or landless young knights who had traveled to war to seek land and fame. This meant (again, in theory) that they had trained together for years and were thus quite used to each other's mannerisms and wholly capable of functioning as a cohesive tactical unit. We have pretty good evidence that they could execute quite complex maneuvers, such as the feigned retreat - not an easy thing to do in the face of the enemy. They generally went into battle in two to three widely spaced ranks, the idea being to give the second and third-most ranks room to maneuver to avoid a downed horse and the first rank room to withdraw in case of repulse or the aforementioned feigned retreat. Knights could and did dismount when the situation called for it - for example, if the ground was unsuited to mounted fighting or to stiffen the infantry. Our sources are far from clear, but my guess is they would remain with their conrois and be posted at the front, where their heavy armor and skill could tell.

We simply don't know very much about infantry organization or recruitment for the earlier part of the period; we know slightly more for the second half of the 12th century. We know that the Normans fielded both heavy infantry and archers in numbers; they might make up 3/4 of the army. The former fought in the usual "western" style, little changed from the last days of Rome, in "close" formations. William of Poitiers reports that they were equipped with mail hauberks (coats) and shields, but I am skeptical as to whether any 11th century western European army could entirely equip its infantry with heavy armor; it may be that, as with the later Scottish schiltrom, only the first rank or two was fully armored. I am unaware of any sources that speak to their organization. It is tempting to assume that they were organized in small units resembling conrois - perhaps the retainers of an individual lord or men from a locality made up a company - but I am hesitant to do so. Assumption makes an ass out of you and me, etc.

Were archers formed into independent units at this point, or were they simply attached to the infantry? It's hard to say; the answer is probably both. Take the battle of Hastings. William of Poitiers, a contemporary chronicler, states that the Normans drew up in three lines, with the archers and crossbowmen at the fore, the armored infantry in the center, and the mounted men in the rear. On the face of it, this seems very strong evidence for the existence of dedicated formations of archers. But the Carmen de Hastingae Proelio, another contemporary account of the battle, tells us there were only two lines: the first of infantry and the second of cavalry. My inclination is to believe William of Poitiers; if for no other reason, it simply makes sense that astute professional soldiers, which the Normans were in spades, would advance with light troops screening their heavy infantry. These archers might still have been ad hoc formations, drawn from the various units of infantry for that purpose; we can't know.

In the late 12th century, Anglo-Norman (Angevin, technically) rulers and lords increasingly made use of a new phenomenon: the independent mercenary company. These common soldiers fought for pay under the leadership of their own captains, but their exact size and composition remains a mystery. It is clear that they could be very substantial forces, small armies, even; but does this represent one large company under a single commander, or an amalgam force composed of several bands? I know you're tired of hearing this, but it's hard to say. Medieval chroniclers were very inexact in their descriptions of them, and frequently refer to them by geographical origin - a band of Brabancons, Welshmen, etc. The validity of this has been questioned recently, and perhaps the chroniclers simply used these terms as a synonym for mercenary, regardless of their actual ethnic composition. These chroniclers generally portray mercenaries as vicious and brutal pillagers, but inferior to traditional knightly armies in open combat. But we must keep in mind that they saw them as being in violation of the societal order, and thus had reason to exaggerate their flaws and under report their virtues.

As /u/MartinGreywolf states, medieval armies were frequently divided into three adhoc sub-armies, called battailes or battles. But we know from the previously mentioned example of Hastings that these battles were not necessarily all-arms formations. In addition to the three ranks of troops, the cavalry at least were subdivided into three additional units: left, right, and center. These were supposedly based on geographic origin: the Normans formed the center and their Breton and Flemish allies made up the flanks.

The structure of the battle was probably the most glaring organizational weakness in medieval armies. As /u/MartinGreywolf says, medieval armies were temporary entities formed from various small units. While the men of a conroi (or, presumably, an infantry company) would know each other intimately, they well might be fighting with strangers on either side of them and under an overall commander they did not know. So, while the troops might be hardened professionals and skillful at the small-unit level, they suffered all the woes that come with an ad hoc and temporary command structure. This was not a phenomenon unique to the Anglo-Normans: professional, standing armies would not even begin to develop until the very end of the medieval period.

Further reading:

France, John. Mercenaries and Paid Men: the Mercenary Identity in the Middle Ages and Western Warfare in the Age of the Crusades

Strickland, Matthew. Anglo-Norman Warfare: Studies in Late Anglo-Saxon and Anglo-Norman Military Organization

Gillingham, John. Richard I

Aforementioned primary sources.