Starts when cold but doesnt start when hot audi a3

could be just a faulty thermostat not opening when engine gets up to temperature, your heater would then blow cold and the engine would run too hot

You may need to reset basic settings for the control module using a scan tool , reading the logged faults stored in the control module is needeed also .

You should be able to cycle the key on and off a number of times to prime the fuel rail. The oil light is always on when you first crank the engine. If it doesnt shut off as soon as the engine starts...yeah, that is good cause for shutting it off and getting a mechanic to check it out.

If you can find the valve on the fuel rail you can check for pressure *(and possibly purge the air out of the rail by pushing the valve and burping off the trapped air)

you could check the temp/fan sensor.but when you say cold do you mean first start of day??cos if the head has hot water in it over a set temp then the fan will start but will go off when cooler.stops hotspots...

If you have no radiator cap that can be removed, you will probably have to do repeated cycles of engine temperature and just add coolant to the overflow until the engine stops draining it. This isn't like a hydraulic (brake) system so the chance of leaving bubbles behind is far less.
With a radiator cap, it's pretty simple; run the engine until the coolant is visibly circulating and add coolant as the level drops.

Check the coldstart sensor,it might think that the engine is still cold and the fuel mixture vill then be wrong,you can also have leaking injectors and then it dripps diesel into the cylinders.

Happened to me too, when the ECO light comes on automaticly (and it only blows hot air) it means it's simply broke.

I don't know the exact production date of your car, but there is/was an airco-related dealer recall for some cars, so just contact your dealer and have it fixed, but don't forget to mention the recall (it might save you a lot of money when your warranty has already expired).

You more than likely have a turbo boost hose leak or vaccum leak, and the burning smell will be the catalyst getting hot from unburned fuel..The car will run very rich with a boost leak.

Sounds like carbon build-up in the cylinders. Take it and have it put on a machine called a "Motor-Vac"

Could be low oil level indicator tha auto shuts of engine if oil goes low