There is pain and tenderness along the inside of the lower tibia (shin). You are probably an overpronator with uncorrected feet and may have begun more hill sessions or speedwork.
A light touch to the bone by your doctor may cause you to overreact! Your injured leg probably looks no different to the other, although running through long-term injury may cause muscle-wasting and alter its appearance.
As above, although x-rays are generally negative and bone scans positive. Biomechanical assessment is readily available, and you should certainly seek it if you have a recurrent or chronic injury.
What else could it be?
All the other lower-leg conditions. There is much overlap between periositis and stress fractures.
Rest is probably the most important element of RICE, as ice itself sometimes causes more pain. Make sure you have appropriate shoes, which will almost certainly include some sort of orthoses to limit your pronation.
If reducing and modifying your training and correcting your biomechanics do not cure the condition, you might need a few non-weight-bearing weeks on crutches, or even an operation called a fasciotomy to relieve the muscle strain.
Can you run through it?/Recovery time
If you have a mild case and make appropriate footwear changes, you may only lose a few days. But if you continue to run uncorrected, and then need surgery, there could well be a three-month gap in your training diary.