Patterns of having bowel movements vary in children, as they do in adults. Constipation is when the period between bowel movements is unusually long and the stool is hard and difficult or painful to pass. A child may withhold stool (not want to have a BM because it hurts), which makes any stool in the colon (large intestine) dry and firm. This results in constipation and an uncomfortable cycle then develops.
Constipation might be caused by not enough fiber in her diet, not drinking enough fluid, stress, a slow digestive tract, or a mild structural change. A serious disorder, such as intestinal obstruction, is very rare and an unlikely cause with your daughter. Fear of using toilets away from home may also lead to constipation. The little piece of skin you describe that hangs from her rectum is probably just a skin tag and not a hemorrhoid. Although hemorrhoids can develop at any age, they are not routinely found in healthy children during this developmental stage.
The best treatment for painful bowel movements is more of what you have been doing. Adding fiber and fluids to her diet regularly (not just when she is constipated) may help. Fiber can be found in fruits, vegetables, and whole grains, like whole wheat, bran, and oatmeal. Children should drink at least four to six glasses of liquids a day, and even more when the weather is warm. A stool softener (like mineral oil taken by mouth) coats the stool, making it easier for BMs to pass. Any little crack (scratch) in the skin where the stool comes out may also be contributing to painful BMs, so you might want to put a little Vaseline on the outside of the rectum. Her sore rash needs to heal some as well. Kids will then tend to experience less pain, so they are less fearful of having a BM. Suppositories usually are not needed regularly. Laxatives and enemas should be used only when recommended by her doctor.
These simple measures work for many children, but it does take time, so try to be patient. If this pattern continues, or you are feeling frustrated, review your daughter's progress with her pediatrician, who may have other suggestions.