Clearly you + party don't want to stick around Beijing for the entire time, even with daytrips. You won't get the "best that China has to offer" in just 9 days. Maybe 9 weeks, or 9 months of travel would be more realistic. So you need to scale your expectations to a manageable small nibbler. More info needed to craft as many efficiencies as possible:

1) On 9/24, what is your arrival flight (Airline, from where, time of arrival).
2) On 10/3, same info as above for departure flight.
3) You mentioned "points" to do the internal China flights. What Alliance--Star or Skyteam? (OneWorld is essentially useless).