I think for car and drivers there is a thread in which drivers and prices were debated in great detail. Or just PM Peter and Moondog and Jie Jie

Before doing the whole rent a car dance (even though time is tight) I would upon landing at PEK, go to someone at an info desk and get them to write out the basic itin (ie. hotel where you are, Mutianyu, airport) and then a line that says, how much? Show it to the cabbie on the way to the hotel and whilst shopping (ie. on the way to the market and back to the hotel pre-wall). If you have no luck, then you can always arrange a rental, though 500 for the day, as quoted here is not terrible by any means.

Would you also consider going further to Simitai/Jinshanling? Normally I would not toss out the idea, but there apparently is a new highway that is now open (I saw it being constructed and nearly finished (with gas stations etc - but no cars), which cuts the travel time from Beijing to about 90 minutes (Mutianyu is 60-90, I believe). Previously, the trip to Jinshanling/Simitai was at least 2.5 hours, though usually 3. Simitai is great since your daughter could take the big zipline back down to the parking lot while flying over a lake/pond. Very cool.

The other question is, where do you want to go shopping? I would try to avoid Wangfujing since there will be nothing really unique there, save for some kitchy junk that will be more than just overpriced. Fake stuff could put you on Silk Street, or for kitschyness and a bit of culture, you could try Nanluoguxiang. That way your daughter and you could get some funky t-shirts/things, walk around a little in a hutong and do something else. Panjiayuan, though to the south of the city is also an idea. Lots of junk and fake "antiques" but its mostly an outdoor market and was the original antique market in the city and could be worth a gander... Get to hotel, relax, swim around?, get dinner and onto the next day, wall and airport.

Essentially depending on where you want to shop should dictate the hotel choice. Nanluoguxuiang - something more on the west side of the city (read: not Chaoyang District/CBD). If Silk or Wangfujing, then you should most certainly stay in Chaoyang or for the latter around Wangfujing.

These are just very general ideas and I wish I could be of more help but my information is a little out of date.