The last time I went up Snowdon the route we took (from a railway station) you would need boots as it was more of a climb than walk,the main route (with the train) running by would be ok with just trainers
But if you got on that train you could wear flip flops and the cafe is very nice at the top
Can't give advice, but my wife suffers from Asthma and has run several races without a problem and her Dad is a 2hr30 Marathon Runner who also has the condition.If you see your GP I am sure you will get some good advice.the main thing is that you are not alone just stay away from the off road stuff (Besides all cross country runners are a bit weird anyway ......)