In summary: cold starting is fine, hot starting is erratic, and the engine runs well once started. If that is correct I suggest there is an electrical or mechanical problem, though that doesn't narrow the field down much. ECM, mass flow sensor, crankshaft sensor, exhaust sensor, fuel injector, and many more could be at fault. I doubt we shall arrive at a firm diagnosis from discussion on this forum, and guessing at answers will be expensive for you, so I suggest you get it tested at an Audi dealer or independant Audi specialist. Do not use on an OBD reader from Ebay or the like, they are unreliable.