I’ll try and answer your questions in order. Firstly terminology: the majority of people living in England at this time period can best be referred to as ‘English’ or ‘the English’. There are good reasons today for naming the people in England before the 12th century as ‘English’ as well, rather than ‘Anglo-Saxon’ or varieties of the term. But there’s an ongoing academic debate about that.

The relationship: it’s worth bearing in mind what led to there being Norman settlers and nobility in England: the Norman Conquest (1066 CE). Duke William of Normandy sailed to England to enforce a claim to the throne of England. So, technically, he was simply seeking to be king. The fact that he brought lots of Norman and other followers with him was to support his military effort. He wanted to reward them, and he also wanted to stay king: he did this by giving followers the lands of major defeated (killed) English nobles. Was this a normal thing to happen in 11th century England? Sort of.

As noted, many Normans were appointed to great estates and smaller holdings. They became the nobility and part of the gentry, and in effect one small group of people at the apex of society were replaced with another. So the relationship between Norman and English is as much one of the powerful and the relatively powerless, as of one ‘ethnic’ group and another.

Right after the Conquest, William retained most of the aristocracy and nobility of England: they stayed English. But the early years of his reign were punctuated by several serious revolts, by English nobles, who wanted to settle scores with each other, or were unhappy with William’s rule (much like later revolts in English history) - there wasn’t necessarily an ‘ethnic’ dimension.

The big episode of violence right after the conquest which is often seen as an example of unhappy English VS incoming Normans is the Harrying of the North. Traditionally it’s been interpreted as a big clash between those two groups. In reality, it was Norman forces hunting down bands of English warriors in remote upland areas of the North who adhered to v small numbers of disaffected English nobles who’d lost power and position under William’s new rule. The story is quite complicated, but that’s the gist of it.

How might we understand the relationship: were the English happy? I doubt they cared very much: most were still paying taxes, might be called up for military service or for labour service. But these were happening before 1066. Might we look for evidence of unhappiness? Well, some pre-Conquest personal names, like Harold, drop out of use: they’re politically too dangerous, even for ordinary folk. Names like Orm and Uchtred, might have a longer life, into the 12th-13th centuries. But in these centuries we also see a lot of Williams, Geoffreys, Roberts: not all of these could be Normans, so some English are adopting aspects of Norman culture. This is just the relationship seen through one body of evidence, but it captures the complexity of how we might understand the relationship.

Anyone left? Were English aristocracy removed? Largely, yes. William wanted to be sure his new kingdom stayed with him, and he saw the best way of achieving this was by having people who he appointed in positions of power, as earls, barons, bishops, abbots.

All of this doesn’t mean that English nobles disappeared. In several cases, including even the family of Harold II Godwinson, daughters of those families married, or were in relationships with, Norman men. It’s not clear these relationships were entirely consensual, which is obviously extremely grim. The best evidence for a continuity of English noble society is down the social hierarchy: the many of small landowners, local ‘gentry’ in England were ‘English’. The picture is very complicated, and across the kingdom is varied: there are fewer English landowners in the south-east, for example, than in Yorkshire.

Over time, in the 12th-13th centuries, Norman and English intermarried, and a new, Anglo-Norman culture and society developed, heavily influenced by Normandy but with some distinctly English aspects.

TL;DR: the relationship was the same as with the nobility and the majority of ordinary folk before the Norman Conquest; there were no great nobles of English extraction left by the 12th-13th century, but mainly because they’d rebelled. By that time, a new society was developing.