Day 1, plan is god, these places are all within walking distance.
Day 2, Take Bus 60 from Nanheyan Stop (南河沿) to Gulou Stop (鼓楼) and then change to Bus 4, getting off at Deshengmen (德胜门) and there take Bus 919慢 to Badaling Great Wall, when getting to Temple of Heaven, you can take Bus 919支 to Jingchanglu Huilongguan(京昌路回龙观) stop and change to Subway Line 13 to Lishuiqiao Station and then change to Subway Line 5 and get off at Tiantan Station.
Day 3, You can take taxi to there since there is no direct bus to Summer Palace from hotel. To Lying Buddha, better take taxi since there is no direct bus going there from Summer Palace.
Day 4, take Bus 103 to Beihai Stop and walk to Beihai Park. There is no place called Chuchuchai in Beijing. To Birds nest, you can take Subway to Olympic Center Station and then walk there. When back to Xidan, you can take subway from Olympic Center Station to Xidan Station, you need to transfer from Ling 8 to Line 10 then Line 4.
Day 5, take subway to Beijing Zoo, take subway line 1 and transfer to Line 4 at Xidan Station, get off at Dongwuyuan Station. You can take subway from Wnagfujing Station to Yonghegong Station, transfer to Line 5 at Dongdan Station. Take subway from Yonghegong Station to Gongyiqiaoxi Station, you need to take Line 5 to Dongdan Station, transfer to Line 1 to Xidan Station and then transfer to Gongyiqiaoxi Station, finally take taxi to World Park. From there you can take taxi to the subway station and then take subway to Wangfujing.

thanks Mr. Michael