Well the first point here is that Henry IV was under considerable pressure from the start of his reign. An increasing number of nobles turned on him after Henry refused to become the lackey they wanted and as he was forced to break promises, and Parliament was strongly against him too as they thought that, since he had used them to take the throne from Richard that he should act as if he owed his very existence to Parliament. This would make it hard for Henry's younger sons to take the throne. Of course Henry had supporters, but with his death likely his opponents would come out of the woodwork. Likely the north of England would fall wholesale to the Earl of Northumberland while Edmund Mortimer would likely rise in the south. Remember that there was the Prophecy of the Six Kings to contend with - a prophecy describing the reigns of the six Kings starting from Henry III and culminating with Henry's reign, it describes him as cursed by God, plagued with scaled skin (Henry had a skin condition) and said that his reign would see England be torn apart by three warring factions, which only in 1404 Owain Glyndwr tried to play upon by signing the Tripartite Indenture with Northumberland and Mortimer (uncle of the child Earl of March Edmund Mortimer) to split England into three parts. People didn't often believe these prophecies, except when they seemed to come true - and the Prophecy of the Six Kings was disturbingly close to the mark. This itself could lead to a catastrophic loss of faith in Henry's dynasty, causing the probable fall of the Lancastrian line. Henry had already suffered four assassination attempts by 1403 and would suffer 4 more, giving an indication of how much the country turned against him. It's unlikely that England would split into three, but the likely conclusion would be Edmund Mortimer, Earl of March being enthroned as Richard II's true heir, and Glyndwr being rewarded with recognition as the independent Prince of Wales. I'd have to say that with the situation as it was, Henry's heirs and supporters would face a virtually impossible task to retain the throne. This war is also almost certain to be over by the time the French "civil war" breaks out, leaving Mortimer available to do all the negotiating.
Henry's younger sons, if they weren't executed as many wanted (to rub out Henry's existence from history) would likely fade into obscurity. Almost certainly their titles, which were removed from Henry by Richard II in 1398, would be again removed for continuity with Richard's reign, rendering them too insignificant to be of use in wars with France.
I doubt the French will want to reopen hostilities despite a resumption of war among the English as the Armagnac and Burgundian parties are happier manoeveuring against each other for the time being though how the English civil war plays out may affect their priorities.
There's war and there's raids. The Duke of Orleans was a war-hawk who repeatedly used his own armies to attack Calais and Gascony, while calling for war on England. There's unlikely to be an actual attack on mainland England but it's entirely possible that Gascony could come under threat even while Burgundy and Orleans faced off. The Dukes of Burgundy also continued to refuse to recognise Henry IV's right to the throne. If for some reason they take issue with Mortimer too (probably not going to happen but it might) then Burgundy might attack Calais with enough force to take it too. Both Dukes had the power to launch their own mini-campaigns on the side while focusing on each other. The intensity of their rivalry, however, means that short of England essentially making a catastrophic mistake and leaving itself open to an attack that France can't afford to avoid, the French are unlikely to declare war. By 1407 though, when the Duke of Orleans is killed, then yes, even the raids on English land will cease as both sides will quickly turn to England for support.
I can still see which ever family is in power an offer from the French of the terms of Bretigny which is more likely to be accepted by the Mortimer/York party than the Lancasters. A Mortimer victory however may set up future problems for England as both Wales and Northumberland-Mann will be more autonomous.
The Armagnacs are quite likely to offer a reversion to the terms of Bretigny - historically they even offered Henry sovereignty of Aquitaine, and they are entirely likely to again here. I'm not sure about an autonomous Northumberland though. With a Mortimer victory Northumberland is likely to fall back into line again. Mortimer, or his regents, will likely re-empower the nobles but I'm not sure about autonomy. It would just be too counter-productive to England's governance, and Parliament is likely to reject the proposal without blinking, and this at a point when Parliament were trying to prove to Henry how much trouble they could cause by flexing their muscle. If anything the main problem for Mortimer is likely going to be that he is going to inherit Henry IV's battle with Parliament over who gets to call the shots.
If the King of England accepts a Bretigny restoration he'll still end up involved in the fight between the Armagnacs (aka Orleanists) and Burgundians. Maybe we even get a division of France into an Orleans France (Kingdom of France) in the North and a Burgundian France (Kingdom of Burgundy?) in the South East, something the HRE may like
I'm not so sure about this. The only side likely to offer Bretigny terms is the Armagnacs, since Gascony was allied at the time to them. The Burgundians were more interested in offering land in the north, and eventually (due mainly to their own independentist streak) the throne to Henry V, and this was a controversial move. But this fight was not originally about the throne. It was about two warring factions of nobles, with a King unable to resolve the problem (due to incapacity). The Kingdom is not likely to split in two, not to mention that stalemates of this kind did not often happen in mediaeval war. If both sides get to a point where they can't defeat the other, the more likely resolution is that both start fighting over control of the King. Since the King can't be split in two, this is not a fight which can be stalemated. Either one side will decisively win, or if Charles dies then the new King of France (which will be neither Burgundy nor Orleans) will resolve the conflict by force. Eventually the scales will tip. Remember that both nobles had their powerbases in the north, and close to Paris. It's all too easy for one of those powerbases to fall, and when that happens the loser will be quicker to sue for peace. Also, the involvement of England is likely to tip the scales. If English troops don't help force the issue then when Parliament tires of the war and stops approving taxes, the withdrawal of English troops will probably do serious damage to whomever they were supporting. Also remember again that the Burgundians and Armagnacs weren't fighting for the throne of France, so they are much more likely to settle for a peace if the war doesn't go their way - they're only fighting over influence, after all.