Can I apply for 144-hour visa free and then 24-hour visa free 8 days later?

I am traveling to 3rd country via Beijing and I plan to stay 31 hours in Beijing, including hotel for 1 night and day trip to Forbidden City. Which option is best to apply - 72-hours visa free or 144-hour visa free? On my way back 8 days later, I will stay in Beijing 14 hours only. Can I apply for 24-hour visa free in this case? I want to book a trip to the Great Wall for this day. Is it OK? One more question - is there a problem if I have stamps from Turkish airports in my passport?