Why did Google and Apple start to design their chips? - And do they operate a chip factory for that? 

Chips have become a competitive technological advantage for the hyper scalers - and no - they do not have their own factory. Today, most chip designs are done using the fabless model where the chip production is outsourced to pure play semiconductor fabrication plants, also known as foundries or fabs. 
TSMC started this model back in 1987 and despite industrial skepticism back then - today are the absolute biggest player with a marked share above 80%.  

Microchips have become critical technologies and are subject to regulation due to the new geopolitical landscape. 
National and regional Chip Acts have been implemented over the last 3 years to ensure sureenety and technology as a competitive advantage. 

How does Denmark fit into this? 
As part of the EU Chips Act 
Our universities doing education and research in the semiconductor area 
Our industry both those who design and produce chips themself and those who use chips designed by others.