So, it looks like we'd have wars in France between rival factions for years to come. Say the English win and secure their rule, how would this affect the rest of Europe?
In OTL the wars between the rival parties in France already had been going on so nothing new there. As for the English ultimate win, I would not bet on it by 3 main reasons:
1st, England was reluctant to spend money on the war in France and an idea of the war supporting itself was not working. In OTL, at the peak of the English successes Daupin (not to mention the rest of his party) had an income which was few times greater than what the English could squeeze out of the French territories they hold. And without money perspectives for raising the new armies were not good (as in OTL).
2nd, while war between the parties made Dauphin and his supporters unpopular in the Northern France, the English were not popular either (why would they with their habit of looting and burning). And while the Burgundian faction was popular, as the OTL demonstrated, its loyalty to the English cause could not be taken for granted.
3rd, tactical advantages of the English military system had been gradually going away when the French started figuring out how to use the firearms. System of gaining by taking the pieces of territory rather than by the risky big battles was working even before.