one of the main differances between pre norman and post norman invasion england was the right to bare arms in saxon england it was the duty of all free men to be armed(according to your means)
Admittedly this Germanic feature lasted more importantly than in continent because of Vikings raids (like it lasted longer among Franks rather than Visigoths, due to the raids from Rhenish or ultra rhenish peoples). I think it would have slowly vanished eventually, while probably letting at least symbolical features.
Egalitarianism did not exsist in that period however the english had more rights and freedoms before the normans.
Again, I put that in question. While feudalism would certainly have at the very least less present in a lasting Anglo-Saxon England, the only certain source close to the PoD (Doomsday Book, who describes England 20 years later when Norman influence couldn't have realistically changed in a radical manner the society) tend to show the presence of group that could be considered as less than continental serves (I'm thinking to cotarii and bordarii, although their statuses are arguably vague and changing)
now back to the actual question,Ireland is often over looked in these questions but given the friendly relations between the ruler of Lienster (Harold godwinson helped the Lienster Irish capture the viking city of Dublin)and the new king of the English could , if Ireland is unified under a single ruler, with English aid then rather then having centuries of bitterness and hatred is there the potential for centuries of friendship and even alliance between England and Ireland?
An Anglo-Saxon England, being more likely turned to North Sea than Atlantic would have indeed less motivation to conquer Ireland than IOTL.
Now, the main issue here is to have an united Ireland. Let's admit that Ardri feature not only last more but prevail. We have still a structure quite close to the original Celtic confederate model that could change by English or continental influence, but without invasion it would take paradoxically more time.
I wonder what are the possibilities of Isles or Scotland to take on Ireland if Anglo-Saxon doesn't go there : Norse-Gaël or Scottish Ulster could be a possibility.
there is the possibility of a more independently minded English Catholicism emerging over time because while Harold would not have been foolish enough to renounce the pope not with the rest of Europe being catholic(at least the parts nearest to England)i could definitely see him (and his bishops)trying to leverage as much as he/they could get out of the pope.
What is a more independent minded English Catholicism? You mean an English church, a more important role of the king in religious matters?
With the rise of pontifical monarchy, I simply don't see that happening.