Los Angeles (Transit via Taipei) to Shanghai Pudong Airport; Shanghai Hongqiao airport to Taipei,

I am planing a trip back to Asia. My itinerary is Los Angeles (Transit via Taipei) to Shanghai Pudong Airport; Shanghai Hongqiao airport to Taipei(Stay for 2 weeks),Taipei back to Los Angeles.
Am I eligible for the 144 hour Visa free? I have two concerns regarding my eligibility. 1. I need to change plane in Taipei. Will this be considered as Taipei to Shanghai then go back to Taipei? or it will be considered as Los Angeles to Shanghai? 2. I will be entering Shanghai from one airport (Pudong) and leaving from another(Hongqiao). Is this allowed?