Hi I would like to spend 3 days in Xian: 1 sightseeing the city and 2 on the huashan mountain

1. To be honest, one day is not enough. You are advised to miss Big & Small Wild Goose Pagoda and Famen Temple. The suggested sightseeing route is the following:
Terracotta Army - Bell & Drum Tower - Dumpling Dinner & Performance
2. You can book the dinner and show in advance on its official website. It has English version that brings convenience for you.
3. Most roads will still open at that time. But some dangerous parts such as Plank Road and Yaozi Fanshen (Sparrow Hawk Turning Its' Body in the Air) will be closed because of the possible bad weather conditions like heavy rain or snow. You are suggested to keep close attention on the news before leaving.
4. As I know, you can still find a tour guide nearby the entrance of the scenic area. But the price would be higher than other season because of the cold weather.

Where can i found reliable information about the weather, the mountain and the route to climb?

Do they give any equipment to climb such as harness and longes karabiner?

The related information on this website are reliable and updated. You can take these for your reference.
And yes, there are several protective facilities and equipments when passing dangerous roads. Don't worry.