What Requirements for stay 1 night in Bejing's hotel in order to connect fly to Taiwan

(1)Re: What documents my father-in-law Need for legally staying 1 night in Beijing Hotel ? The Round trips:
Washington D.C. USA ---Beijing, China--- Taipei, Taiwan. // Taiwan-- Beijing,-- Washington D.C. USA.
My father-in-law's a "U.S. Permanent resident" and has "Taiwan Passport". He's scheduled to fly Air China
from Washington D.C., USA to Beijing, China. He needs to stay 1 night in the hotel in Beijing in order to
Connect flight in the next day to Taiwan. On his return From Taiwan back to Beijing, he needs to stay 1 night in hotel inBeijing to connect flight back to D.C. USA.

(2)Re: What documents my mother-in-law needs for legally staying 1 night in Beijing Hotel for connecting
the next day flight to Taiwan ? Is she qualified 72 hours Transit Without Visa Program ?
My mother-in-law's a "US citizen and has USA Passport". Her Round Trips' flight schedules are the Same with my father-in-law.

(3)How about on their Return Trip Back to D.C. USA from Taiwan.... Do both of them Need to apply 72 hour TWVP in Beijing airport because they need to stay 1 night in Beijing airport in order to connect flight back
to D.C. USA ?

Truly thank you for your Quick responses of my questions. Help ! Help!
Ueh