I'm facing mostly the same problems described here.
I started running very recently, and after 2 months I started feeling this right hip pain, or discomfort. The strange thing is that I felt it not while running but the day after.
Went to the doctor, made some x-ray, and he said nothing was wrong with my hip or joints. So I decided to visit a physio who, after seeing my x-ray, said that I have 2 things:
FAI - Femoral Acetabulum Impingement
Weakness of the stabilizing muscles (gluteus medius is the main one)
So I went through a lot of sessions and exercises at home, and 2 months after stopped running, even though the pain was not completely vanished, I restarted running slowly.
But the pain came up again, and the strangest thing is that I can stand the pain while running, and even sometimes the pain is not present when running. But it simply comes the days after, and specially when I'm sitting at the office or while driving my car.
I really don't know what to do. The pain comes and goes. Increases and decreases intensity. Some days I think I'm ok, and others I really feel the pain while sitting.