Will there be any kind of agreement in the latest scenario?
- Harald's in control of Northumbria and consolidating control of the rest of "the North" - somewhere between the green and black lines
- William has "the South" and is expanding above the red line
In all honestly, again, redline would show "maximalist" advance of Normans.
At worst, I can see William only on Godwinson lands, arguing that he defeated him and therefore have the right to take the spoils
At best, he would advance roughly up to the red line, leaving former Tostig lands and the lands of Morcar, that was already attacked by Tostig and Norwegians at this points, and that probably expanded on former Tostig lands as, he's accounted to have warred with Welsh (something you need an actual common border with to do so).
And that's assuming that everything goes for Normans' wishes. Again, we're talking of a Norman army after a battle (hence with losses) and of a fresh Norwegian-Saxon army that likelt didn't encountered serious resistance. Guess who'll have the upper hand?
Do they armistice?
One attack the other?
Temporary truce?
It's much probably they had an agreement on the principle to sharing the island, as it was already done IOTL. So all that matter is to more or less settle the share, make a formal oath, and...that's all.
Do both maintain claim to Kingship of England?
You don't split titles, so yes.
I probably haven't been explaining myself too well
1. Without an oath then both go for as much of England as they can grab and hold.
You don't have only two choices : oath that bind everyone, and non-oath that allow random stuff to happen.
Again, as Tostig went first in Norman court, then in Harald's, it's pretty safe that an agreement was reached at least on the principle (and it may be what decided Harald to undergo the whole expedition)
2. Edwin and Morcar are of no use to Harald unless he wants rid of Tostig(who was nearly as unpopular in Northumbria as his elder brother Sweyn had been in Mercia) but are of use to William to forment trouble and slow Harald down.
I didn't suggested he made friends with them, I suggested he conquers the hell out of them.
The impopularity is also irrelevant. Normans were also impopular, it didn't stop them to crush the hell out of whoever didn't liked their rule.
And for slowing down : I doubt, I very doubt that a William currently stuck in the south up to a Hastings-like battle would have the material possibilities to pull a Machiavel there, at least up to he controls at least the South (as in, first control what you want before tring to support someone else).
As for the capacity of resistance of Edwin and Morcar, as Harold massed a same army IOTL for both Stamford Bridge and Hastings, it's pretty safe to assume that it would be limited, maybe a thousand or two against a whole expedition of at least seven thousands.
3.What happens depends on how badly mauled the Norman army was at Senlache. The better condition it is in the more land goes to William and the less likely he is to just accept a division of England.
Again, between a fresh, untouched army; and an army that if Hastings serves as plausibility measure, that have relative serious losses...I know which I'll pick.
Everything doesn't depends on William's will, and he would probably understand that a foe with Saxon allies (something that Normans didn't) have a better legitimacy and capacity to have local support than he does. He can bid for Godwinson lands and maybe some more, but unless William is on steroïds, he won't go immediatly for a fight.
(After he secured his grasp on Southern England, probably, but not immediatly)
4. Both will claim the title of King of England but William is more likely to be accepted by the French,HRE and Papacy.
Why? You assume that only one king would be recognized. It's much more probable than both will be (Middle-Ages England wasn't didn't have One-China policy)
France : As the Capetians were fighint William and Normans since decades, I don't see why they would be more interested on William being king (and a more serious threat) than Harald.
Papacy : Unless we give credit to the tale about papal support on William's expedition, I don't see why.