I recommend the Redwall Hotel.
It is close to the back door, Shenwu Gate, south gate of Jingshan Park and south gate of Beihai Park.
Take subway Line 1 to Xidan from Tian'anmen west of Tiananmen East in front of the Forbidden City. Change to Line 4 at Xidan and get off at Beigongmen which is the north entrance to Summer Palace. If you want to get to Badaling, take subway line 4 back the city center and alight at Xihimen. Change to Line 2 and get off at the next stop: Jishuitan. Get out from Exit A and walk eastward to find Deshengmen Gate. Tourist bus 919 will take you to Badaling. Remember the 919 bus station is just next to an old city gate - Deshengmen. Do not get in other 919 buses. It is getting cold in late October, but still warm. Need to wear long-sleeve T-shirt and take a warmer jacket to proof wind. I suggest you do not visit Summer Palace and Badaling in one day. It is a rush.

can i go to mutianyu great wall and then still visit the ming tomb? are the two places too far from each other?
and can we just rent a van? how much will it cost and what number do we call...

Thanks Mr.Qi for the response.

Follow up question, The direction you've given, Is that from summer palace to badaling?

Im planning to go to forbidden city & summer palace on day 1, the day 2, ill go to great wall. Can u pls tell me how to get to the hotel you suggested to badaling? how much would it cost for ride? how many hours would it take? thanks a lot.