I really like some of your questions. It is true that there is no concrete definition of configuration space, in keeping with the Scientific American Library horror of anything that looks like a standard math book presentation. My plan was to illustrate the concept by a bunch of examples and hope that a common notion would emerge. The whole idea is that we have a collection of things that can be specified by a certain number of coordinates, the "controls". If there is just one control, like a slider bar or a dial, that determines the thing, then we say we have a one dimensional space of configurations. A thermostat or the volume control on a radio are examples, and it is easy to come up with many more.
But there are many situations where the outcome depends on more than one variable, and that leads to higher-dimensional configuration spaces. I generally think of a configuration as something spatial, like a geometric figure. In one of the Java applets I have been studying, there are various commands that make it possible to define points in the plane, line segments, and polygons. Each type of object has its dimension. Points require two numbers. So do segments on the x-axis. Segments parallel to the x-axis form a three-dimensional configuration space, since it takes one number to specify the location of the horizontal line, then two more to give the endpoints of the segment. The configuration space of all segments is already a good example of a four-dimensional space. I'll talk about this and other examples in class today (Friday).
It is true that when we have a configuration space like the positions of the humerus and the ulna, then restricting the motion of one of them lowers the total dimension, and that is like projecting something from 6-space down to 3-space. The nature of the projection depends to some extent on the manner in which the configurations are described, and part of the art of this business is to choose descriptions that relate one kind of configuration space to another where we already have a good hold on the mathematics. That is where linear algebra and topology come in, but those are two other stories.
The question of where things really exist is, as you point out, different from the number of dimensions required to describe them. Anything that exists in one dimension automatically exists in higher dimensions although that doesn't necessarily mean we have access to them. Flatland is a part of Spaceland, but A Square can't get out of his plane on his own. With help can we ever attain a position out of our space? It isn't easy to answer that, and properly speaking it is not a mathematical question, more of the domain of physics, or psychology, or religion, or a mixture of the three. Several authors have probed these areas, P. D. Ouspenskii being one of the favorites of some of my former students.