Huashan in December

Yes, it is open in winter, but it will be closed if there is snow.
For reaching there, you need to climb to reach. Or you can take the cable car to reach the north peak and then climb there.
From Xian, you can take direct tourims bus line 1 to reach there directly or take the bullet trains to reach North Huashan Station. Then take taxi to reach.
According to the previous experience, you cannot finish the visit in half a day. You can spend half a day to reach the mountain and then climb it at night and go down in the morning.

yes,it is open,yes,if you take cable,it can be done i half a day