constant sky

I know there are probably plenty of game programming sites that have information I am looking for but just in case the answer is short and simple:

It looks to me like many games render outdoor scenes by attaching some sort of texture map (representing the sky) to the clipping planes. It “sort of” looks like the objects disapear over the horizon. Is that right, is that what is being done, a texture map attached to the clipping planes?
If so, why do it that way? Why not push the far clipping plane back so far that objects can be seen until they become smaller than a pixel? Will that distort closer objects?

The closer are near and far clipping planes, the more accurate will be the depth buffer, avoiding ugly artifacts like sawtooth intersections.
Again: objects clipped are not drawn >> more speed with nearer far plane (less objects in the viewing frustum).