Thanks for the reply, Jim
Yes, the starter cranks fine, the engine just won't start. So a "bad main relay" could be the problem, as you say.
I'm just wondering what it is about this 5-10 minute window. It probably has to do with something getting too hot maybe, and needs time to cool down?
As simple as you make "check for fuel and spark" sound, I must admit I'm too car-stoopid to know how to handle that. I assume it means checking for blockage in the fuel line somewhere, and making sure the spark plugs are working, but still, I wouldn't know where to start.

h:
If something needs to be replaced, I'll have to have someone else do it. I just wanted to have an idea what I was in for before I took the car in for service.