Terrionne....Yes you can just make sure you use some good tape.
It would be better to use a vinyl Christmas tablecloth if the door is not sheltered by the elements.
If an interior door, wrapping paper would be just fine, any kind of wrapping paper, if exterior, you'll need something that can withstand the weather. Just tape it to the inside of the door.
For the exterior, maybe a Christmas themed vinyl tablecolth-as Heje suggested, or and alternative could be a shower curtain, depending on the size of your door.
The dollar stores should also be carrying door covers by now. You could also check Amazon.
Yes! I'd make sure to use the thicker/heavier duty wrapping paper (Hobby Lobby has great wrapping paper!). It may fade a bit if that side of the house gets a lot of sun, but I think it'd be ok for one season.
This may help: http://homeagainjiggetyjig.com/wp/more-christmas-wrap-decor You may need to cover the paper with packing tape to make it more weather proof