I was just researching no/low heat for the A3, and a known issue is sludge or some such in the heater hoses. I cannot say if it is an issue in the A4 system, but since you are gonna deep dive it will be worth having a look to see if the flush cleaned them out enough. Issues also exist on the A3 (and perhaps the A4?) with the air flow doors (or the motors that operate them). On the A3 I turned on the recycled air switch on the climate control panel and warm air started to blow (for now

). There is a diagnostic section in the FSM titled, D3E800C3A16-Heating__Ventilation_and_Air_Conditioning.pdf
If the white smoke is concerning then keep an eye out for oil level, milky oil (coolant contaminated), or oily coolant in the reservoir. If you'd like to be more certain then get a leak down tester (Harbor Freight sells a cheap one that is adequate). There are plenty of videos that cover doing the leak down test (differs from a compression test - the engine doesn't move during the test), but basically you are looking for overall pressure holding power of the assembly and differentials between cylinders. But there is a lot more you can learn during a leak down test by listening to various parts of the engine. Use a length of fuel or vacuum hose as a mechanic's stethoscope - open and listen down the dipstick tube and you will probably hear a high pitched hiss. That's air escaping from the cylinder into the crankcase, and a little bit (past the rings) is normal. Low pitch, or what sounds like a lot of air is bad. Next, remove the air duct at the throttle body, open the throttle body and stick the 'stethoscope' in there. If the cylinder is indeed at TDC then it should be quiet. If there is hiss then an intake valve is leaking or burnt. Checking the exhaust valves for leaks is a bit of a pain - you can remove the pre-cat O2 sensor to get the hose in a listenable spot, but even then it's tough to hear. That's all cool, but the reason I'm suggesting the leak down tester here is for your concern over the head gasket. A leaking head gasket will reveal itself during the leak down test as either bubbles in the coolant reservoir or air escaping into another cylinder (listen through spark plug hole). Also, if your spark plugs seem super clean you might be getting coolant in the combustion chamber. One last thing - compression can be lost through a cracked block as well - either into another cylinder or into the crankcase.
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