Either one should work as long as you can hang it securely.

Yes, just make sure the curtain rod extends beyond the edges of the door. You might want to add a weight at the bottom (sew a picket and insert a dowel).

Black out curtains are awsome. They will most likely be lighter weight than a blanket. You can get them inexpesively at Bed Bath and Beyond.

Yes, definitely it will be a good way to stop hallway light. You can use any other dark color curtain also for this.

What about weather-stripping around the door frame?

When I read @dabbler answer I realized I had read the question too fast. I would just roll up a towel for now and put it in at the bottom of the door. Eventually you can put weather stripping as he suggested.

Also, you might want to try wearing a mask to sleep. The little bit of light in the room can be good if you get up in the middle of the night.

Lastly, I thought I would mention if your door opens into your room it will be a pain with the curtain. Usually doors open into a room rather than out to a hallway, but not always.