Many parts of the Great Wall in Beijing can be visited except the Simatai under renovations. Badaling is the closest to the downtown area, but more touristy than Mutianyu. If you want to stay one night in Mutianyu, there are only some homestays - living in local villager and room rate is cheap. You can take bus 936 at Dongzhimen to Mutianyu. Generally speaking, bus 936 may stop running directly to Mutianyu since Nov 25. So, if they don't going to the great wall, you can get off at Huairou bus station and then change a minibus.