First of all - your hotel is in a great location!
Day 1 sounds good. Mutianyu section is less crowded than Badaling section. The stairs are a bit steep at some places, but there's a rail next to the stairs all the way up... You can start your day early and get back to Beijing in the afternoon, have some lunch and then head to Summer Palace. There's a lot to see there and a lot to walk if you want to see everything, because it's a huge place, but if you walk at a nice pace then it could be a nice stroll for the evening.
Day 2 - you don't need a car for this day. I agree with Jean-Pierre D: it's better that you start your trip at Temple of Heaven. Wake up early and see all the locals doing exercise and playing all kinds of games. If you like shopping or just want to get impressed by the huge Pearl Market that's full of everything you can imagine (not only pearls), then you can exit Temple of Heaven from the east gate and walk just a bit to the market's entrance. Then get on the TIANTANDONGMEN metro line (5) that's situated right next to the east gate and at Chongwenmen metro station change to line 2. Get off at Qianmen station. Then you can get out of the metro (the whole ride will probably cost you around 2 yuan) and start walking north, through Qianmen street (which is a nice street that was reconstructed before the 2008 Olympics and now you can find here lots of stores and interesting side streets. I would recommend eating lunch in one of the side streets before you head to Forbidden City). At the end of Qianmen street you'll find Tia Sq, where you can stroll around for a few minutes. Then you can enter the Fobidden City. There's a lot to see there, but I would advise to stay there for no more than 4 hours. Check the Forbidden City's website before going and see what you really really want to see there. End your visit at the north gate. Then head back to your hotel by foot or hire a rickshaw (if you find one that will take you for a reasonable price).
Day 3 - Jingshan Park is very close to your hotel, so you can start your day there, but you won't miss anything if you'll skip the park. I would rather visit Beihai Park, personally. You can walk to the Bell and Drum Tower, yes. It's not along walk. Then you can walk all the way to Lama Temple (although I would rather visit Confucius Temple, which is right next to the Lama Temple) or you can walk north from Drum Tower until you hit Guloudajie metro station (line 2) and take the metro to Yonghegong station, which is situated right in the Lama Temple. The huge building that you will probably see behind you is the temple. If you have time, you can visit this temple and Confucius Temple, which is in a side street next to Lama Tample (check map). I really enjoyed Confucius Temple and learned a lot from it. Then you can get on line 5 of the metro station and switch at Dongsi station to line 6. Get off at NanLuoGuXiao station, which is about 10-15 minutes walk from your hotel. You can hire a car for this day, but I think you can get around easily with the metro or veen with taxis.