Depends. Sibling rivalry varies tremendously with the ages of the children, the characteristics of the family, the activities in which they engage, and the children themselves. Some interesting studies have described more rivalry with children who are closer in age and the same gender, but this depends very much on the individual children involved. Can it be as bad in girls as in boys? In a word, yes.
Maybe? I don't know if it's a matter of "should, " but brothers do not have a monopoly on sibling rivalry. Rivalry between sisters and sisters, sisters and brothers as well as brothers and brothers is common.
Should they? . No they shouldn't, but they do. Neither should brothers, but they do. We call it sibling rivalry. It can last a life time.