John would live longer. Though I'm not sure whether to have him killed in battle or captured by Louis' forces. If he was captured, would they just keep him imprisoned or execute him or ransom him? What might the effects be of each?
The execution is really unlikely. The medieval use was far more the imprisonment for moral and religious purposes, as well for keeping a valuable hostage against family or supporters.
Ransom, in this precise case, seems to me unlikely. And, first, who would pay it. With which treasury, as the one of John is likely to be captured if himself is taken?
I think we have two solutions : or John is imprisoned and let in his jail until he died (sort of Trencavel fate), or he's released when the situation improve for Louis and he could benefit from a demesne in France (not in England, too close from his old power and eventual supporters).
In the latter case, it should be far from Normandy and Poitou. Maybe Gascony, in order to make the south of Gaul more troubled.
If he's killed in battle...Likely as OTL, the barons would likely prefer the suzerainty of a child rather than a powerful king. Another good reason to treat John well and to keep him alive.
Now that's interesting. Could you tell me any more about this status quo? Or anything more about the situation in southern France at the time?
In 1216-1217, you have the occitan revolt against the crusaders. Raimon VII enter back in many of his cities formerly taken by Simon de Montfort and this one didn't managed to take them back.
It is not really desesperate for the crusaders, but if it things goes like OTL, Monfort would die during the siege of Tolosa and his son being proverbially unskilled ("being Amauri") and without royal help (as Louis would probably busy with England at least up to the 1220's) the reconquest of lost desmenes by occitan lords is almost certain.
For the situation before the crusade, you had occitan demesnes that recognized the french king as a distant suzerain (helping him when they didn't had another choice), critically because the huge demesne of the Plantagenet was a threat (with possible inheritance from Ramnulfid claim on Tolosa). Apart from that, they were autonomous or even independents with their own vassals being autonomous from them.
Of these possibilities, I think the second is probably the most interesting, followed by the third. But could you tell me any more about what events might lead to these different possibilities, and what effects they will have?
1. Louis VIII accept the demand of vassalage of Raimond VII, contrary to OTL.
2. Louis VIII refuses, and Raimond demand the king of Aragon to be his suzerain as he proposed to Peire II in 1212.
3. Both Louis VIII and Jaume I refuses. This last one could be because Jaume was under the tutelage of templars (it can be ignored, as willing nobles can take care of this obstacle) or because Raimond was the main artisan of the occitano-aragonese defeat of Mureth.
Therefore, by being a good penitent, he could say a big sorry to the pope and offer his lands in vassalage. It could work, even if the Pope would claim predominance on certain lands (maybe Provence with Beucaire, and possibly the half of Lower Languedoc).
If Louis isn't able to keep North England, what happens there? Does Henry set up a government in exile there? Or can it become part of Scotland? Is it feasible for Henry to make a government in exile in Ireland?
Henry is too young for setting up anything but tower of cards. You could have the same situation than Occitania at the end : nobles acknowledging from far a suzerainty, maybe Scottish or English.
For Ireland...No. Ireland is likely to be shared in little normano-irlandese demesnes, and don't have the background to serve as anything ending by -in exile. I'm not even sure this could exist in the XIII century.
If Henry is not with his father, he would likely join the court (or being more or less captured, more accuratly) by an ambitious noble opposed to Louis. The more likely is he's taken with his father, or quickly after and as Jaume I, being under the tutelage of Louis or a french noble.
Ah, but how might those Barons react to this? And I assume Louis (and his successors) would rule England and France as separate kingdoms. But I guess they'd bring in lots of French influence (even more than in OTL).
This is the little problem : French kings didn't have appeal for "separate kingdoms". If his sucessors manage to keep the two thrones as long (let's say 3 generations, maybe 4) the two crowns would be quickly merged.
It could admitedly take longer, without the influence of meridional scholars that existed OTL thanks to the conquest of Occitania, but not that much and the exemple of Bolonia's school is just too important to be ignored.
For the french influence, I would say the norman model of feudality and law would likely been replaced in the south of England by a mix between norman system and french custom system influenced by the Roman Law (by that I mean the interpretation of Justinian Law that appeared in Italy).
The inheritance question, both in France and England came far later. It wasn't a rule before the HYW when each side claimed opposite stances in order to inherit the throne. Let's admit the female inheritance is possible in this TL as this war would be butterflied, and considering the role of queens (both consorts and mothers) in the court of France.