England begins his royal numeration after the Conquest (William the Conqueror, 1066). That's why Edward I was not the first Edward, for there were three other Edwards (Edward the Confessor among them) ruling pre-Conquest England.
But yes, the numeration goes regardless the dynasty. But it seems like the number is a matter of naming the king, and if the king is made from a republican president or a Lord Protector, he just does not have the name. You can always modify this, I usually do. Tired of kings named Louis and Philippe in a Burgundy game, I decided to "fake" (in IN, not in HttT) a dynasty change: the house of Valois-Burgundy fell and the House of Lorraine got to power (at the same time, I annexed my vassal Lorraine). So I changed the history file to add royal names like "Thibaut", "Simon", "Eudes" and "Robert". It turned out that Thibaut III was a great king, and Robert V converted to protestantism.
In this case the Duke of Malborough may hate the old dynasty and is forging a new one.
Yes, but that's not how it worked. Any king in the XVth Century would number himself "as it is proper". The Swedish and Norwegian kings count also mythical kings that never existed, thus the last Eric was Eric XIV, but only eight Erics before him are historical. But Eric XIV though that he needed to represent those legendary kings into the numeration.
Another example: Peter the Cerimonious, king of Aragon, signed as
Pere Terç, Peter III, even though he was the fourth Peter to rule over Aragon. He did so because in Catalonia only had been two Peters before him; Peter I of Aragon was not ruler of Catalonia, but Peter II was, after the Crown of Aragon was created.