One should really keep in mind that feudalism works on a pyramid scheme. The vassal swears fealty to the liege, promising support in exchange for protection. If a liege does not deliver that protection, the fealty of all his remaining vassals will probably take a serious hit. This includes protecting one vassal from another. So it's generally a bad idea to wage a war if you are a small fry, because most of your potential victims will either be stronger than you, or have a liege who is.
However! Sometimes your potential victim's liege (which may be your own) will favour you, so that if you pretend your war is due to some just cause, he'll rule in your favour and that's it. Alternatively, he may be busy with another war, or simply too weak due to age or infirmity or whatever; and if you are bold enough, you may then snipe at his vassals... but it's risky, because in one year or five or thirty someone may take that aggression of yours as justification for grabbing all your land (though of course, if you then have a strong liege who favours you, your attacker will think twice about it).
If your opponent admits defeat, and signs a peace treaty, that's it. If your troops defeat all opposition, and manage to collect (and keep collecting) taxation from the local folk, that's it again -- but it does not have as much legitimacy, so you'll probably want to have some higher authority recognize your claim to the land (which you should probably want to do anyway: as mentioned before, if you are a small fry without a protector, very soon someone will try to gobble you up).
In early 13th century, just deciding to conquer a neighbour was a bad idea - unless that neighbour was of the wrong faith and had no treaties with your lords and lieges. Yes, feudal structure was fragile and prone to turnovers: but for this very reason people on all levels of society generally stuck to privileges, rights, duties etc., and not doing so will soon turn somebody into an outlaw.
Typically the first step to conquer some territory within a feudal society is finding some title to do so. Marriages, deaths and following inheritances or stewardships, claims by your lieges and such can all lead to such titles, which often are disputed and hence reason for feuds, wars and conquest.
If two of his vassals have a conflict about some land, usually they ask their liege to decide, or contend in the courts of the relevant duchy, kingdom or empire. Fights among them might flare up if one vassal bent on conquest tries to grasp a contended resource (typically a castle) from another while the conflict is still decided at the respective court: but then the liege, duke, king or emperor will assert his authority to have the fighting stopped, and will try to impose his ruling about the contended resource.
Of course, lieges also die, and while their inheritance isn't yet settled their candidate heirs may look for allies among their vassals. At such times vassals looking for chances of conquest take sides: and fights between vassals can occur, that are only thinly disguised as support for respective candidate heirs. While this is happening and greatly weakening the fiefs disputed among the candidate heirs, the courts of dukes, kings, emperors will be very busy containing the strife and revoking rights of those fiefs.
And then dukes, kings and emperors also die, go to war or on crusade, usually leaving temporary administrators to act in their stead who might be particularly approachable to favoritism, bribe, blackmail, threats ...
I think you understand the concept now, and see how all but those immediately benefiting from strife were very much in favor of settling all consequences of deaths, inheritances and marriages years before they might happen. While most expected their neighbours to fall upon them in case there was any doubt, and prepared for the worst.
To imbed all of this into a campaign, best look up the mundane history of the areas around your covenant. Making up credible conflicts from whole cloth by using general descriptions of feudal society as a base is IMO far more challenging.
Although I agree with much of what you say, I'd add one other element: civil war. In England the wars against the various kings are so endemic the person you are most likely to be fighting, most years, is the person on the other side of the civil war, and often regions split as the leader of a region goes for one side, and their chief rival goes for the other. After the war is over, the winners tend to be rewarded from lands taken from the rebels.
Formally might be a bad idea. Pyramid system and all that. True? Only formally
I recently read a book about the Catalan counts and basically they spent more time fighting each other than the moors to their south. Most of the time they were cousins claiming that this or that village or castle belonged to them. Generally the church (that generally were relatives as well) stepped in to BEAT THEM UP instead of preventing violence. So, as long as you have a claim to a land, you can go and kick the neighbor. If you have no claim, invent one and raid for "resources that he obviously owed you".
Generally things did not escalate to full oiut wars, but a village or 2 along with their strong points could change hands. After a generation or 2, another party (maybe the former owner) would recover them. Some castles in Catalonia have had over 40 owners in 3 centuries.
The best things to wage war for were the multiplicity of allods scattered along the territory.
The feudal system was much more patchwork than the ideal picture shows. Conflicting allegiances were in the order of the day. You could easily be waging war against a vassal of count A and B (he is mostly a vassal of count A, but you wage war for a land that he holds for count B). If count B is in unfiendly terms with count A, count A might not look with bad eyes to your actions. Both of you could be vassals of count A in this sense. Count A will not interfere since this would weaken count B despite meaning a war between of his own vassals.
Just remembered another one. During the reign of king stephen (weak king in England) the father of William Marshall (a minor baron) waged war on hos pown interests on nearby counts. He even got William captured as a hostage for his good behaviour, proceeded to kick an opponent in the gonads (and capture his castle) and claim that he still have "the tools to make more Williams" in case the hostage was killed. He kept doing that and switching allegiance between Stephen and Maud (even if he chiefly supported Maud) to justify his attacks until he was finally run down and killed after a battle lost by Maud's supporters IIRC. His death for keeping his allegiance was used as an example of what he should have NOT done and as a puzzling case for his contemporaries, that assumed that he would surrender and switch allegiance again instead of allowing himself to be killed covering Maud's escape.
There is no way to sufficiently describe the actual workings of feuds resulting from territorial conflicts in early 13th century western and central Europe in an RPG forum post. The best one can do is describe fair weather generalities and hint at some of the most likely exceptions - the rest is for study of the SG, who knows time and place and interest of the players.
I think I understood, that the campaign of Heaven's Thunder Hammer is placed at the lower Rhine near Cologne. So Marc Bloch's La Société Féodale (Vol. II) (old, but groundbreaking and widely available in translation) would cover the basics for him - but that is already a huge book.
Often it's not even that they got rewarded, they just ended up with someone's land after it was all over, and nobody was prepared to kick them off it. Also bear in mind that civil wars aren't soccer matches, you can change sides even if nobody blows a whistle. Well-timed treachery, I mean realising who the true King was, could get you what you want.
In case of contested dukedoms, kingships or such, almost every vassal and noble maneuvred to end up on the winning side, and milk the conflict for what it was worth. Not doing so meant neglecting the care of your own subjects and family, after all.
As an example from early 13th century Reich (hence of interest for the campaign of Heaven's Thunder Hammer), Walther von der Vogelweide in his König Friedrichston addresses very openly, how he shifted allegiance from Otto IV to the more generous Friedrich II and obtained a fief from the latter. Of course, Walther was an author and propagandist, not a conqueror in civil war. Also not fighting within the Reich, but French and Papal money, the battle of Bouvines and realistic politics decided Friedrich's acceptance as German king.