And do I use tap water to wash the cooling system or waste a lot of distilled water?
Yep. I use tap water and change it every couple years anyway. But my tap water is probably much different that yours.
Another question is, I know I fill it with water and run the car with the heat on max until it warms up, and after that drain all the water, this also helps wash the system, but how to fully drain the system, so only the premixed cooling liquid is filled, I thought it will be done with compressed air?
See comment below about heat setting. Don't waste your time. The only reason to run the engine is to use the water pump to push the coolant around, thus evenly mixing what's in the head, block and radiator. Until you're ready to do hot system pressure testing (look for leaks).
Coolant system cleaning solutions help clean the system - hot water does not. Although hot water is a good rinse cycle. It's VERY important to get all the cleaning solution out before installing the semi-permanent coolant. And be sure to rinse several times.
For draining/rinsing I took off the ~1/2" heater hoses at the thermostat housing and connected the garden hose there. Filled until it flowed out the lower/upper radiator hoses, stopped and let it all drain. Repeat until perfectly clear. Then drain as much as possible (which does not drain the block).
There's a coolant hose going from the block to the turbo that would allow *some* draining of the block to occur but it's a real PITA to access. Looks like the support brace on the turbo inlet side has to come out and it's still a cramped space. Best to replace that hose also as it's probably old and brittle by now.
Using compressed air to push out the coolant would be very difficult at best. It's not like there's an access port at the top and a drain port at the bottom. You could probably blow out each tube and hose - if each end is freed - but that's not gonna work for the head/block/water pump/turbo unit as a plumbed assembly.
Make sure you get all the bubbles out and turn up some heat at full blast.
Heat setting does nothing to help here (although it *might* slow down the temp rise of the coolant). Coolant is already flowing full-time through the heater core - all the heat setting does is move the air flapper valve in the AC mixer box so more air flows through the heater core instead of around it. BTW - ALL air flows through the AC evaporator core no matter what the heat setting is.
BTW - I haven't seen a coolant flow control valve in an automotive application since the mid-70's.
BTW2 - Since many of these cars are 5+ years old, it's time to replace all those coolant hoses too. Plan to make a weekend of it. And get the proper set of clamp pliers - THAT alone makes this job so much easier.
I posted all the hose the part numbers previously, iirc.