Business trip to Hong Kong

Yes, you can, but under the following conditions:

- Your flight from Hong Kong to Qingdao has to be a nonstop one
- Your flight from Qingdao to Europe has to be a nonstop one
- You are not allowed to have any additional stops within mainland China of any kind

If the above mentioned conditions are fulfilled, you are definitely eligible for 144 hour visa free transit in Qingdao and you don't need any kind of visa. Of course, one of the conditions is that your citizenship is among 53 eligible countries. 72 hours visa free transit has been replaced in Qingdao by 144 hour visa free transit a time ago.

Also, you have mentioned that you need a visa for Hong Kong. If you are a citizen of some European country, you don't need any kind of visa to visit Hong Kong.

I have a Travel document for Refugees under the un convention of 1951 (The passport is blue). It is issued in Denmark.
I just called and they told me that I cannot enjoy the Visa Free transit. What about if I am flying to Hong kong on the 16th/01. Travel back to Qingdao in the morning on the 18th/01 and on the same day, at noon continue to take my flight to Europe (Non-stop in the mainland). In short, Can I connect my flight so that I don't really have to register that I am entering China on the 18th/01?
My flight to Europe is already purchased and cannot be changed to be from Hong Kong.

Thanks in advance!

In reality, there are no flights through China where you will just transit unregistered. This is simply not possible because even if you are only in transit, you will have to clear some, at least, a shortened form of immigration.

Anyway, this should not be a problem if you are using just 24 hour visa free transit through mainland China and if Qingdao is your only stop you will be definitely eligible for 24 hour visa free transit and you don't need any kind of visa.

You will print out your flight tickets and have those printouts with you. Also, you will declare at check-in in Hong Kong that you will travel to Europe via Qingdao using 24 hour visa free transit. After a short inspection of your tickets, you will be allowed to board the plane.

Upon your landing in Qingdao, you will immediately approach 24 hour visa free transit because you will almost certainly travel on separate tickets. You will be allowed to collect your luggage and to perform a recheck for your flight to Europe.

In short, this is it :)