Separate bookings/airlines for 144 TWOV at Beijing

There are no problems when in possession of separate tickets. Since the very first days of visa free transit, a long time ago, separate tickets were allowed and this is true until this very day.

Your itinerary is definitely eligible for 144 hour visa free transit in Beijing area and you don't need any kind of visa.

You should just print out your flight tickets and have those printouts with you. Also, you are advised to arrive at check-in in Chiang Mai a bit earlier because you could encounter less informed and less trained staff there. If necessary, you will kindly tell them to consult Timatic in case of their poor understanding of 144 hour visa free transit and if necessary, you will ask for a superior. In any case, just arrive earlier at the airport to catch your flight on time and with peace of mind.

What if I change my plans like this:

Instead of flying to Beijing from Chiang Mai I would fly on 11th January from Kuala Lumpur to Beijing via Hong Kong (one itinerary, transit only in HK) on CX. As I understand HKG is not considered as Chinese airport in this regard so would this qualify?

Yes, you have understood this subject perfectly. Your new itinerary:

Kuala Lumpur - Hong Kong - Beijing - Helsinki - Stockholm

Is definitely eligible for 144 hour visa free transit in Beijing area.