Aye, generally speaking three things can happen; your feet can become un-flat (Such is the case with me!), you'll get on with no problems in spite of having flat feet, or you'll go ahead and end up buggering up your knees with all the running.
Just have your GP take a look and see what they think.
Depends on the type of flat fleet. Mine are nearly completely flat and I had to and see a consultant to see if they were spasticated (seriously, that is the term), which essentially means rigid. Mine were not so I was fine to join.
I have had some dramas and find that my feet bruise easily as a flat foot doesn't transfer impact as well as a foot with a pronounced arch. However, a set of sorbothane insoles does wonders. The army also had a set of custom made insoles constructed for me, which are excellent. I've ran for the army so it doesn't seem to have done me too much harm.
Best of luck and, as Wayner says, don't take no for an answer. Particularly from some civvi Dr who doesn't understand the issues.