I think you should go to whichever doctor you can get for the least money. Even if you go to a non-sports doctor, they should be able to see if you have shin splints even if they call them by another name.
Anyway, there's lots of things you can do for your shin splints. One, make sure you're getting calcium, even if it sounds stupid. Trust me, it really helps them get better and makes them hurt less. You could also look into getting the standard orthopedic things you put in your shoe at basic shoe stores. (Dr. Scholl's, etc).
It's bad to run on concrete too. Try and avoid that if you can.
Have you tried doing an ice massage after running to help reduce the inflammation? I don't know what a doctor will do to help you with the shin splints, as it is a muscular condition. Be sure to stretch out the calves and do some strengthening exercises. One that I have done is to write the alphabet with each foot while seated.
for the kidney stones, you may want to look at your diet. It could be lack of magnesium (Which also helps your muscles to relax).