I just saw the mention of kids' ages on your Sanya thread. That changes my suggestion for Beijing a bit. I would still engage a private car for at least part of the day, if not the entire day. Trying to maneuver three kids (2 of which will be small enough to get tired well before day's end) on subway and public transport is not a great idea. Technically there is a limit of 4 passengers allowed in a taxi but since your 5th is a 2-year old "lap child", you might get away with it. I don't think it will be as comfortable as private car or small van, though--especially for the segments to/from Mutianyu. With 4 people (I think 2-yr old is free) on the Airport Express each paying RMB 25, you might as well be in a car or taxi for a flat fee on the return journey.

These days, a decent car with driver for the day for 8 hours-ish should be about USD $100-120 and a van about $150. To take care of all your transport needs with no worries, might be worth it. A half-day is probably about 60-65% of the full day price. For the first part of your day from the airport hotel, I wouldn't deal with the taxi in this situation, arrange a car ahead of time. One of our FT'ers recently used www.tour-beijing.com strictly for van+driver rental (her party paid for their own entries and "guided" themselves at each sight) and reported back favorably. Look for posts by worldiswide. You will not save that much money by taking a taxi to Mutianyu instead, and again, if you are not a Mandarin speaker you will have to figure out how to deal with communication issues as 99% of BJ taxi drivers simply do not speak English (or anything else).

I think the Mutianyu GW and the Forbidden City on the same day is really going to be too much for the two youngest children. I seriously doubt they would appreciate the FC. At the risk of being a BJ sightseeing heretic, after the GW, I'd blow off the FC entirely and go to Beihai Park which is not only historic, but has paddle boats to rent and just is more engaging for kids. Go in the south gate and out the north gate. It's a very pleasant place for the whole family, and you'll see other Chinese families with kids there. After exiting the north gate of Beihai, cross the main street over to Houhai area. I normally don't recommend adults do the bicycle pedicab hutong tours in favor of hiking/biking oneself, but by mid-afternoon your kids will need to rest their legs. I think they and you would find the pedicab tour through the old hutongs a hoot--you should be able to put 1 adult + 1 kid in pedicab A and 1 adult + 2 smaller kids in pedicab B. Drivers will do the pedalling, you just sit back and take it all in. After that, you should be sufficiently refreshed to find a place in Houhai to have dinner--lots of restaurants there and nearly all are reasonably priced. If you decided to keep the driver for the entire day, a meeting point will have to be arranged and a time--that's the downside of having the driver for the full day. If you prearranged to release the driver after a Beihai Park drop off, then you can get a taxi back from Houhai. I'd just have the taxi go all the way to the hotel, forego subways and light rail. Show the taxi driver a card from your hotel and you should have no problems.

I think this is a better plan to balance seeing some classic Beijing sights for the adults, yet keep the interest of the kids in the family.

ETA: what time of year are you coming?--I think the Beihai paddle boats don't operate again until April or maybe May 1. The sights themselves and the hutong pedicabs will always be running, though.