I did wait 10-15 min on idle with turned on seat heat and low blower control. I would say that right at the beginning I saw a faint white steam. But, after a minute or two the steam was very very faint. In addition, since the car was standing a couple of days I assumed that it's normal. Did you see the video in my previous answer? Is it normal?
My other question: when I saw overheating indicator I stopped immediately, waited 15min for engine to cool down, filled another 1/4 gallon and continued driving slowly back to home. No overheating indicator was on though the way back to home was even longer (10 min). Could it be that the thermostat opened and let the coolant go through and now that's fine? I just want to understand maybe now it's fine since the coolant level didn't go down.

 

Do exactly as stated, start your Mini up and let it idle for 15 minutes, get your phone out and start a timer seriously! Then closely monitor the engine bay (underneath also) during the warm up period of 15 minutes, if you do not see or hear steam sizzling around the thermostat housing and under the engine bay/exhaust, drive it only around your neighborhood no faster than 20 mph and have extra coolant or distilled water with you just in case. Stay away from traffic lights that could potentially prevent you from pulling over safely and I want you to pay attention to any white exhaust smoke that may exit your exhaust during the warm-up period of 15 minutes and while driving!

Check your coolant level every few minutes (seriously!) so that if you start losing coolant you can catch it before it gets to low and you can pull over to the side of the road and turn off your engine and add more coolant or distilled water once your engine has had time to cool down. I suggest you not leave your immediate neighborhood meaning to stay off the surface streets. Your neighbor's might look at you funny but it beats buying a new engine.

 

I will do what you suggested as soon as I get home (by train). I'll appreciate if you could answer the following 3 questions:
1 - Since I don't know what is bad amount of white steam, I need some kind of reference/threshold. I know that when car stands a couple of days it might have a white steam. The video that I've posted - is it normal amount of white steam?
2 - I want to ask about the bleeding bolt. It was written in many places that after starting the engine on I should pay attention to coolant tank. If I see any bubbles I should open the bleeding bolt and let the air come out and keep it open until I see coolant start coming out of the bleeding bolt. So, I did open the bleeding bolt, waited but no coolant came out so I assumed that the whole air came out. Is it ok that no coolant came out from the bleeding bolt when it was open? Does it mean that there is no air?
3 - If I'll hear sizzling - does it mean that there is still air in the system or what? What should I do if I'll hear sizzling but no leaking anywhere?

 

Wow....
Do yourself a favor...
Get the car to a shop...you are in way over your head...
You will never be sure, and could be making things worse...
A doctor, no matter how well trained does not perform his first survey solo...
And you are hoping to do car repair without basic foundational knowledge (the strength of a bolt or nut can be guessed by the diameter of the shaft, and tighter is not better, mating surfaces on parts, new and old must be prepped or they will leak, etc). Not you fault, but sometimes you have to admit you have been beaten. Many of have learn the basics from helping a parent growing up, doing stuff with friends, etc...forums can be a big help, but it is not a substitute for the basics....
Please don't construe this as anything more than a reality check, but it sounds like very little has gone right so far....and the questions you ask only reinforces my feelings that you need more help than a forum can provide to prevent serious financial losses....

 

If your bleed screw was open and coolant did not come out there is air in the system, coolant must come out if there is no air in the cooling system. If you have to ask those questions you should not be performing this type of repair period. Take it to a shop before you permanently damage your Mini Cooper. You're in way over your head and lack basic automotive knowledge. Let us know what the mechanic says.

For now it's just a simple diagnostic fee versus serious damage if you continue with this on your own.

 

Ok.
But, still can you please answer me: Can I still get the air out? I will warm up the engine on idle and then open the bleed screw and wait till the coolant comes out and then close it?

 

So while you're burning coolant through your exhaust (white steam) and hearing a sizzling sound, so now you want to bleed the air out of your cooling system?

You need to listen to reason and take your Mini into a shop before you brick your Mini Cooper, I'm not going to help you destroy your Mini Cooper.

 

But a few hours ago you suggested by yourself to try again - "let it idle for 15min with phone". That's what I'm doing.

 

I didn't see your video in your previous post of the white smoke coming from your exhaust, that should alarm you! You didn't read the mixing instructions on the gal of coolant, you asked for the torque specs for the thermostat housing and ignored it because you didn't even use a torque wrench causing a bolt to snap in half, nothing has gone right so far for you. Things will only get worse if you don't take your Mini to a shop! You're ignoring advice given to you by myself and ZippyNH, why bother giving advice when it is ignored?

 

I'm checking the closest good repair shop for diagnose. What should they check? They need to take the head apart to check the head gasket or what? Just asking so I'll know if they making up unnecessary job or not.
By the way I checked the oil and its color and viscosity is like oil and not too liquid or white. It could be a good sign, right?

 

Systemlord, Though it took longer than I expected (I had relatives visited us and needed to find and buy some specific screw drivers) I fixed my car more than a week ago and just wanted to update you guys. I drove my car, on weekend, to a repair shop near me for diagnose for $20. The issue was that when I returned one of the pipes it was loose causing the coolant to drop down and getting eventually into the exhaust pipe. I drove my car back to home and tightened the pipe. Everything is fine now.
Anyway, I liked the experience. Learned a lot.

 

Wow, that simple and here we thought the worst.

 

Good to hear it was a simple, cheap fix!!

 

So what is the vin number of this mini so nobody ever buys it used from some poor guy who gets it next?

 

Amen brother. Every step of this process had me on edge almost biting my nails. As Systemlord wrote: why offer clear and precise instructions when each time they are ignored or not understood?

Each post from the OP gave every indication that nothing went right and everything was done wrong. Sheesh. Don't do this to us again Theatist. I just stumbled upon this today because Zanydr added a comment a few days ago.