Air in cooling system after flush W140

[majooo]

Hey everyone,
this is a great forum, it helped me several times before.
I've searched this forum for answer, but could not find anything relevant, so I am posting here for my first time.

I have 1997 S500 and previous owner used green antifreeze fluid.
I've looked in my Star DVD and it said to remove drain plug on radiator and two drain plugs on block, which I've accomplished.
The tricky part is with refill:
On the DVD it just says fill the overflow tank and worm the car with cup removed, easy.
When taking car for a spin, no heat at all. Than I've heared a lot of bubbles behind the dash - heat come.
I've checked the overflow, but it was OK (no light on dash came on neither).
After a 3 miles car was at 120 and no heat. I let it cool off and was able to put another gallon in.
Still a little bubble noise heat was weak.
When I got home I start looking around the engine bay and found "mystery" hose, that goes from top of the overflow above fan and curves back to the termostat housing (cca 0.5 inch diameter).
There is a T connection on this hose with a small hose (6in) heading forward (next to the air intake). However there was screw at the end of the 6in...
I've removed this screw and stem start to come out T.
I've let it cool off.
Finally stem was gone and I put over a gallon in the expansion tank, until water started to come out of the small T hose. I could hear bubbles
Is this T hose designed for getting air out of the cooling system?
Remove the screw or keep in place?
How to get rid of all the air trapped in the system for good?

I'll be doing the flush several times, so it will be a good practice...

Thank you in advance for all your ideas and help in this matter.
M#.