All right, your character looks pretty good so far and you have a good idea on where you would like to go with her, which is always good, so to answer your questions:
1) Nobles in Egyptian society were related to the pharaoh, priests, scribes, doctors, lawyers, or important military personnel. Many of the nobles were overseers of the lands worked by peasants. So, in short, they were the government. Your character's father would have most likely been the Pharaoh's relative or the sibling of another important government person and his job would have probably been to oversee the peasant workers (unless he was a government person himself). The rich in Egypt lived in a separate part of town from the peasants, often protected by walls, their homes were large, having as many as 70 rooms for both the family and servants to live and work in, and often had their own courtyards in the middle.
2) Although men and women in Egypt had traditionally distinct powers in society, it seems that there was no insurmountable barrier in front of those who wanted to deviate from this pattern. Egyptian society recognized women not as equal to men, but as having an essential complentarity, expressed especially in the action of producing children.
The info for the second answer was taken from
this page, it's a wikipedia page on women in ancient Egypt and you may want to take a look at the "Working Women" section because it has some information on the roles women played in both the upper and lower classes.
As for how the father would have treated his daughter we don't really have a lot of information but it would be fair to assume that he would give her very little freedom in her youth and going out into the city, unless it was to the bazaar with her mother to shop, would be nearly impossible without having to sneak out of the house without being seen by servants or guards.