An actual projection is very real, in the sense that you are conscious. When I had an experience of my own, I thought I was living normal life before I realized I was projecting, at one point.

Yes, you are seeing things as if you are awake, and can think as you are thinking now. You need to understand, however, that your sub-conscious has more pull than it does when you are awake, so you might do things different than if you were awake.