Is this takeoff of a 767 in bad weather conditions a normal procedure?
Take-off within the crosswind limits is a normal procedure. From the video it's hard to tell what the crosswind was in this particular situation, so it is impossible to tell about the legitimacy of this take-off. 

Rotation seems to be on the 1000ft marker, i.e. 1000 ft from the runway threshold. But in case of a displaced threshold the take-off distance available may be more than 1000ft from that point. It was airborne before the end of the runway.