Knighthood had not become a distinct institution at the time of the initial Norman settlement. Knighthood basically coalesced during the period of the 10th and early 11th centuries, simultaneous with the development of the duchy of Normandy.

Not to my knowledge, no. After 150 or more years, the Norman aristocracy, besides being thoroughly culturally assimilated, was hopelessly interbred with the native population. Additionally, it should be noted that most aristocratic families of the 12th century, as illustrated by Duby's study of the Maconnais, could scarcely trace their ancestry more than a few generations.

It's very difficult to say with certainty; what can be said is that the so-called classical method of training knights (a formal arrangement of a period of time spent as a page, followed by a period of time as a squire in service to a knight) only applied to the Late Middle Ages, if then. I can speak to the 12th century, but we really don't know as much as we would like about the earlier period of knighthood. An aristocrat might train his son, but if he were poor or undistinguished, and wished him to have a better martial education and to form better connections, he might be sent to the household of a greater lord, usually one whom the father was connected to by marriage, blood, or bond. This seems to have generally occurred around early adolescence - perhaps age twelve or thirteen. There he would have done arms training with the knights of the lord's household, learned to ride and hunt, the rituals of court and feast. Once in adulthood, he would continue to train, either with the knights of his lord's household, or, if he became landed, with the other knights of the neighborhood, who in war formed the smallest tactical unit (1-2 dozen) called a conroi.

I don't subscribe to the view that Norman knights were innately better than those of other regions of the Frankish world. Where the Normans really stand out is the degree to which they exported military manpower abroad, and their propensity for expanding to fill any power vacuum. They produced copious numbers of younger sons with no prospect of inheritance, and unleashed them, armed and hungry for land or at least subsistence, on any region of the world in need of mercenaries. Norman mercenaries were brought into southern Italy in the mid-11th century and, before the century was through, had supplanted the local rulers and were reigning over half of the peninsula. Even before William's invasion of England, Normans had been brought over as mercenaries or military advisers, perhaps in an attempt to create a Saxon cavalry tradition.