Is it necessary to wear furred boots in Xian? Any idea if it will snow? Will be heading there from 13 - 19 Dec.

It is necessary especially you came from Singapore. It will be chilly in that city then.
It is very easy to snow in December, so you may meet the snow. But the exact situation is hard to tell and it is suggested pay attention to the weather forecast in advance to check the snow days.

Thank you.