Is it normal to put the Christmas tree up on the 5th December and take it down on the 5th January?

"Normal" is all relative. Some people, who have fake trees, can leave their decorations up from October until March or April. Some people buy cut trees, which dry out and die within a couple of weeks, so they don't put them up until at least mid-December, and take them down in early January. Some people do not decorate at all.