Wow, so many places. I am only able to arrange some of them which I have been to.

Tiananmen, Forbidden City and Beihai Park are close to each other, you may put them into one day. Xidan is also not far, but seems time will be not enough.

Prince Gongs Mansion and Houhai are not far from each other. BTW, there are Hutongs around them. There are many bars in houhai area too.

Go to the Great Wall in the morning and then the Olympic stadium. The night scene there is very beautiful.

You can take subway line4 to Summer Palace, old summer Palace and Beijing Zoo. These are close.

Fragrant garden and Summer palace can be covered in an evening... It is a wonderful experience to climb to top of hill in Fragrant gardens. Do not take the usual concrete path while climbing up, prefer trekking up the hill. Come down by the cable car...

Olympic stadium closes after the evening, please confirm the timings before you go. Go in such a time that you can go in, and by the time you come out, its night and lights are on...

Don't waste much time on palaces and temples... You would find all the architectures repetitive...

Out of your list.. I think must visit Places are: Tianmen & Forbidden city, Behai park (one evening), Olympic Stadium, Great Wall, Summer Palace, Beijing Zoo (Dont miss Dolphin show), Temple of Heaven, World Park...

For Shopping, I also suggest you to also visit Dashilan street & Pearls Market (includes toy market) for shopping.. Also, don't miss the zhonguancun (HaiLong) electronics market.

Hi,as I like shopping,I will advise your shop plan.Xidan and Wangfujing is so famous,but you may just choose one of the two palce,and XIdan is more popular,Wangfujing is for some high-end.but if you choose wangfujing ,I think you could to to GUOMAO to see the urbanization of CHina.Have fun!