Can I stay in an ordinary hotel with a residence permit in Taiwan Province Province?

Of course. According to the Catalogue of Valid Hotel Identity Documents, the identity documents of China citizens (including China citizens among Hong Kong, Macao and Taiwan residents) mainly include: resident ID card, temporary resident ID card, driver's license, residence permit, passport and seaman's card; Hong Kong and Macao residents travel to and from the Mainland; People's Republic of China (PRC), mainland travel permit for taiwan residents and other tourist cards.