Flight route check for Transit visa

My itinerary is as follows -

UK-Amsterdam- Shanghai- Taipei (I have around 70 hours in Shanghai)
Return - Taipei-Beijing-Paris- UK (I have around 60 hours in Beijing)

My questions are -
Is Taiwan definitely classed as a 3rd country for the purpose of free transit ?
Does the fact that I fly from and to my home nation (UK) via another country have any impact on using the free transit ?

thanks
Apologies-posted this on wrong forum- now added to 144 hr Transit forum.