People with flat feet can definitely have pain in their feet and legs with intense physical activity, including running cross-country. However, the same can be true for people with particularly high arches. In general, a person's foot structure is one of those things like eye color or hair color or skin type that is simply unique from person to person. The most important thing is to make sure that you have the right shoe support for whatever type of feet you have so that you can participate in activity without pain.
With regard to foot support, a podiatrist
can often be the most helpful medical specialist. He or she can examine your feet to determine exactly what degree of arch support you have. In addition, podiatrists can also create orthotics which can be worn inside of your running shoes to help either compensate for the lack of an arch or support a particularly high arch. If your foot/knee/leg pain associated with running is due to improper arch support, orthotics can be very beneficial.
However, some of the joint pain
you describe in your knees and other leg muscles after running may not necessarily be associated with your feet. For this reason, it may be helpful for you to see an orthopedic surgeon
for a more detailed evaluation. Running can cause a host of orthopedic injuries, especially in young women, and having a physician evaluate your symptoms and physical exam may help identify conditions that could benefit from physical therapy.