what's a good location to stay at Beijing?

Oh, you may stay in a hotel near Qianmen or Wangfujing so that the transportation to the mentioned spots can be convenient.

I don't think it is a good choice to take a taxi to tour around. On the one hand, it costs much more than buses and subway; on the other hand, traffic ham easily happens in Beijing. Considering all, I'd recommend you take subway, which can lead to almost all mentioned attractions expect Great Wall. And bus is suggested for the Great Wall if you are not in a rush.

Since you have four days, you can also visit the Wangfujing Pedestrian Street, Yonghe Lamastery, Bird Nest & Water Cube, and Silk Street. One day tour to Tianjin is also worthy!