I think cutting Chengdu make sense. You can do Tianmen Square Forbidden City, maybe visit Jingshan and Houhai and then Temple of Heaven in one day. Save Summer Palace and Mutianyu for another day.
I recommend you take taxi to/from airport, price is cheap and subway isn't very luggage friendly, especially at certain transfer stations.
Ah ok. Thanks. So for the bigger transfers after initially getting to the city it's best just to get a taxi to take us to/from the hotel (would you suggest this as well for Xian and Shanghai too?). And then just use the subway for things like Beijing Zoo, Summer Palace, etc. and generally getting around (where it's not easy or too far to walk to) once we're settled. I would still try to look for a hotel that's located near or within walking distance to a metro.
For Great Wall, is it easiest just to arrange for a private guide/driver to take us to Mutianyu direct from the hotel?