I'ill be travelling to Beijing and Xian in mid-december and visit the great wall and huashan.

The temperature in Xian differs from 30F to 45F and Beijing can be a little colder with the average low temperature of 25F. It sometimes snows.
As I know, Great Wall sections and Huashan are still open during that time. But before you go to Huashan, you'd better check the weather in advance for it is closed in bad weather conditions.