China Passport with US permanent resident

1) I know someone that has China Passport and a US Permanent Resident Card (US Green Card). They are transiting/transferring in Taipei to Hong Kong. They will stay in Hong Kong for a few days and then enter Mainland China. Will they need any visa for this transit/transfer?

2) They will eventually travel within China to Shanghai. They plan on flying out of Shanghai to Taipei and transiting/transferring to the US. Will they need any entry permit for this transit/transfer?