This is a bit of a tricky question to answer, but rather than just link back to the numerous 'Feudalism doesn't exist' threads which you can find in the FAQs (and I do encourage you to have a look at them, because there's lots of interesting and relevant stuff there, even if it is a little bit more complicated than it first appears), I'm going to give it a shot. Fair warning: this answer is more about why we can't answer this question and why people keep asking it anyway than any real 'this is what feudalism was'.

So firstly, we can't really answer this question because, as people who've spent enough time on this sub will probably know, the 'feudal system' didn't exist. However, that little aphorism doesn't really tell the whole story - what we need to start saying is that 'The "system" didn't exist.' There was no system which governed every aspect of life in the Middle Ages, as the 'feudal system' makes out. I'd go even further and say that there was no system which governed even easily systematized spheres like government or the military for large parts of the Middle Ages, although that's certainly much more debatable.

Now to be clear here, I'm not saying that the medieval period was a lawless, anarchic, free-for-all without any government or social control (although some historians (Hi Tom!) have argued this for certain times and places). What I'm trying to say is that there was no over-arching, coherent or consistent system which regulated and delineated legitimate power and authority. Power and authority in the middle ages were constantly being claimed, denied and negotiated. What I mean by this is that power was contingent, not absolute; how powerful you were depended not on your office or title, but on your political surroundings and context (which is why it's very common to see, particularly in the central Middle Ages, people simply give themselves the best title they can get away with - the counts of Rouen -> the dukes of Normandy is a classic case).

[As an aside, the previous comments on the contingency of power are hugely generalised, and they won't apply equally everywhere throughout the entire Middle Ages. In particular, there's an increasing trend towards the formalization of government and institutions in the later middle ages (from the mid-twelfth century onwards) which counteracts some of this, but never comes even close to eliminating it (I'd argue it's still very present even in our modern, hyper-formalized and systematized governments). Byzantium and the Arabic world, which both held onto late Roman exemplars of government much longer than the Western Christian world, are also obvious cases where things are quite different.]

What this means for medieval Europe (and I'll exclude Byzantium here, because it's totally different and I don't know enough to comment knowledgeably) is that there was no single system (or even multiple systems) which united politics, the military world, economics and social control. It's not to say that it wasn't organized, but it was organized in a messy, ad-hoc way - based on deals, compromises and short-term contingencies, not any long-term or systematic plan.

Which brings us back to the Crusader Kings (or Total War, or Age of Empires, or [insert medieval-based strategy game of choice here]) - why do they persist with 'feudal systems' in gameplay? Put simply, it's a game. You need a coherent and consistent system to make it fun and fair to play. No-one wants to play a game where your big invasion of Germania is suddenly halted because you have to go broker negotiations between two lords in your army who refuse to fight until their disputes over woodland access are resolved. Because so many people's first introductions to the medieval world are through video-games (and I count myself among those), it's very difficult to get out of the mindset of 'well, there must be some sort of system'. Combined with how government/civics is taught in schools (where the focus is on the constitution/system of your country - how MPs/congresspeople/Presidents are elected, what the checks and balances are, how various institutions have their spheres of authority - rather than how politics actually works), it's no wonder people have trouble grasping how the medieval world functioned without any of that structure - and so scholars in the past invented systems like 'feudalism', which have continued in the popular imagination for exactly this reason. As a result, we have questions like this, which are so totally grounded in the assumption that there must be some kind of system that they're quite difficult to actually answer - but I will give some indication of how some of the issues raised in the question fit into this picture of a 'messy' middle ages:

1. Like I said before, I'm going to leave Byzantium out of it, because I don't know enough about it. The big difference between England (Britain is anachronistic in this period) and France in the central Middle Ages is that the king was far more dominant in England than in France. This is, in my opinion, largely a result of the king's massive personal wealth in England combined with the fact that the conquest of 1066 allowed the top tier of English landowners to be replaced with families who had close personal loyalties to the royal family. France, in contrast, was more 'egalitarian' - there were many aristocratic families who could realistically challenge the king in terms of wealth and raw power, if not ideological authority. This created very different dynamics of power and politics in the two kingdoms.

2. I'm not really a military historian (maybe u/Rittermeister can give more details), but as a general rule I think it's better to think of medieval armies as more analogous to modern urban gang wars than to a modern military. Armies consisted of aristocratic followers and their retinues, who would fight when called upon (subject, as always, to negotiation and compromise over pay, duration of service etc. etc.) - they would be self-equipped and organized, but well-trained and professional (or as well-trained and professional as you'd get in the central medieval period). Mercenaries were also a key part of most armies - untrained peasant levies coerced into combat are by-and-large another video-game myth.

I realise this is slightly all-over-the-place, but I hope it helps!