Mightn't Edward the Exile face hostility from the Godwinsons, who don't want to be eclipsed/displaced by a blood-relative of the king? There are theories Harold got rid of him IOTL.
Eventually, you might have tensions, but outright hostilities seems hard to reach this quickly.
In late 1050's, Godwinsons seems to have enjoyed royal favour, especially with Leofricsons being the official rebelious earls for now; if Edward is credited with an ealrdom (which isn't that obvious : I don't remember heirs recieving such in the XIth century), it would likely not be directed against Godwinsons (altough you might see this being switched, depending on the situation, in the 1060's).
As for outright murder, I don't know : it seems to fairly ignore the nature of late Anglo-Saxon kingship, and the half-secular/half-saralized nature of the House of Wessex : the succession was expected to be dealt with by the earls and the Witen, but it almost always involved a Wessex king or being related to (a good part of the absence of overwelming support for Harold, IMO, comes from this)
Or, what would have happened if Edmund Ironside had not died after his agreement with Canute? Would Danish rule last in that circumstance?
It's kind of another situation than described by the OP, but if I can pull my two cents.
"North Sea empires" as Canute' tended to crumble easily under their own weight : each regional nobility and elite had its own interets and cultural organisation, and the royal hegemon as it was established wasn't popular even in Scandinavia, mostly due to ressource focus.
Basically, every part of such hegemonies ended to rebel, forcing the king to take ressources to crush it, that in turn made nobles of other regions rebelling, due to what was percieved as an unbalance.
it doesn't help that scandinavian succesion, at this point, was less a matter of dynastic succession than kingship succession, dosn't help : if your predecessor was both king of Danemark and England, even if you weren't related to him and elected in Danemark, you still had a legit claim on England), you'd have much room for infighting, even with an agreed split.
Giving that a lot of late Anglo-Saxon identity was built on the stress they were not Scandinavians, you'd be likely to see at least several attempts (at best after Canute's death) to join up with Wessex. Either Danes realize that and attempt to crush Edmund even if he lives, either they don't or can't, and they remain with a knife on their soft insular underbelly.
I'm not sure Edward has the political capital to bring the Godwins under his influence.
It more or less depends how the 60's unfolds : you could see an important Norman, but quite possibly German as well, influence that would be just enough to balance a bit Godwins' influence.
You might see Godwins to be relatively okay with Edward as a king : not only a legitimate king in late AS England was to be the source of political success (contrary to what happened in the continent, he still had the monopoly on the honores and benefices distribution), but the lack of much viable alternative (at this point, Edgar was far too young tobe one) IOTL might advert how the political crisis unfold. At least for a time, Edward would be seen by the great houses as a way to both instaure some balance against the others, or at the very least, prevent one to win it all.
My question is more cultural based, how much of what we recognize as 'English' culture would emerge under these circumstances, even with strong Norman/French influences, if the rulers of England are still as Anglo-Saxon as their peasants then how much will change compared to RL?
Well, the idea that Anglo-Norman elite was distinct from their population is real, but I think it was pushed to extremes for some nationalistic/quasi-romanticist reasons in the XIXth, and somehow went into popular conceptions.
Truth is, the cultural differenciation wasn't an exception in Middle-Ages, unless arguing that the culture of Capetians was the same than Francilian peasants, or that Latin principalties provoked a latinisation of Palestine.
While Franco-Norman nobility in England was certainly more differenciated (would it be only because they not only got rid of Anglo-Saxon nobility after a while, but also because they represented an extremely reduced part of the overall population, compared to continental nobility), it didn't that affected popular culture and language in the Middle-Ages (so far, I was under the impression we did used a western Germanic language on this board)
You had a tendency, since the IXth century, to translate into continental terms the English situtation, a tendency that was accelerated in 1066 for obvious reasons : but institutionally, the continental influence (whatever Norman or Ottonian) isn't going to leave any time soon, and even without Norman invsion, an equivalent to Davidian Revolution in England is to be expected.
Some of the direct legacy of Normans, as French-issued words and names, would certainly be a miss in such TL, but institutionally, you'd still have (eventually) adoption of quasi-feudal models, encastellment, etc. Arguably you'd probably end up with specific institutions, making England probably looking more like IOTL medieval Scotland.
So, certainly changed, but not litterally unreckognizable England or English medieval culture, IMO.
I've often seen it argued that some form of democratic culture might emerge sooner, due to the presence of the Witan
It's essentially a protochronic ior culturo-centric form, tough : because the English medieval Parliment eventually evolved into a first form of parlementarism (which really became democratic only quite late), not every proto-Parlementarian or Parlementarian feature was going to.
We could as well argue that Gothic practice of Councils since the VIth (and you'd have more of a case making these institutionalized assemblies) was going to involve an earlier democracy; or that Anglo-Saxon England was more likely to devellop a plantation economy because it relied more than its neighbour on slavery.
The Witan was, mostly at this point, an formalized happenance of a concept that was really important into royal politics in Middle-Ages, namely advising the king (as the earlier form of royal councils in France, altough it was much more complex with the appearance of other institutions as Estates).
It was basically a gathering of the political elite, as you found elsewhere (and had few similarities with the supposedly egalitarian* germanic councils), which was non-permanent, non-institutional and proteiform.
*Another tarte à la crême of early middle ages historiography.