Before I setup OSM I need to choose between host the planet or just a section of it. I would prefer the first one, but I'm afraid of the effect that this will have.
Is there an option to get only a few countries on my machine, and the request for other titles being relayed to other server? Or they will just not appear on my site?
You can host the planet or just some countries, it's possible. The manual how to setup a OSM server: http://switch2osm.org/serving-tiles/building-a-tile-server-from-packages/
It's only depend on your server capacity.
I don't know of any ready-to-use solution. Because a rendered map usually consists of square tiles you would have to define some rather complex comparisons for each single tile to decide whether to retrieve it from your own server or a different one.
