If it will be run inside a wall cavity, you will need to have it hardwired correctly. If it is in a drop ceiling to an existing outlet, I think a long power cord would be ok. I would get a 16 gauge, 18 is a bit thin.
Pulling from the ceiling fan involves the possibility that someone can come along and turn OFF the ceiling fan right in the middle of something you're watching. Plus, with some projectors, that's not good to do that.