When I wake up during night, I sometimes have double vision for a little while. Is this normal?

It certainly can be. A lot of people have a "phoria": the inability to keep the 2 eyes looking together when they are tired. This is quite common, and usually lasts for only a few moments. Phorias do not need to be treated. If you experience double vision at random times of the day or the double vision does not go away, you should see an eye doctor. You may want to see one anyway if only to reassure you.
Usually normal. While asleep, our eyes cross slightly outwards. This is because the bony orbits supporting our eyes are positioned outwards and while awake, our brains must make an effort to align the eyes so they "point" forward. Upon awakening, one may be aware of this "adjusting" period and perceive two images until the eyes are aligned.