arrive in Ningbo airport (from Taipei) and leave from Shanghai airport (to Hong Kong)

Ningbo is in Zhejiang province, so you don't need a transit visa.

That's not allowed as Ningbo is not one of the four approved airports (which only include the two in Shanghai, one in Nanjing and one in Hangzhou). I've also read elsewhere that some had been subject to hours-long interrogation to get 24-hour transit and that was because they know someone who lives in the province and had Chinese visas in the past. If you decide to go this route you may be sent back to HK.

What you can do is to go to Ningbo after clearing immigration in one of the four approved airports, the two cruise ports in Shanghai, or the railway station in Shanghai. You cannot enter from Ningbo to benefit from 144-hour transit.