Wireless Internet: Well ... almost

Well ... almost

Your suggestions are mostly right, except for the placement of the routers.
While logically it would seem that it makes sense to put them closer, that is not always the case.
Think of the wall in between the two and the path that the radio waves have to take, keeping in mind that the waves travel in a straight line.
So, while it might make sense to put the two routers closer physically, one must ALSO insure that the path between the two is "straight" through the floor and walls, else, the 6" floor can become 24" and what is gained by the physical proximity is lost by the fact that the waves must travel through MUCH thicker walls.
As for the original question ... WDS is your friend. (Of course, it also depends what hardware bits you already have, and how much you are willing to spend. I have hacked together this kind of setup in a variety of other ways - for example, game adaptor and wired router.