This could be a very long thread, for several reasons. Witten http://www.sns.ias.edu/ckfinder/userfiles/files/%5B52%5DProc_Shelter_Is_II_1983.pdf: "As soon as one begins to think about Kaluza-Klein theory, one faces a bewildering variety of choices. There are many assumptions one might make, and many facts about elementary particle physics one might try to explain." (And he said that before modern string theory, with its googolesque landscape of vacua, was developed!) To this I would add that decades of study of KK theories has produced a vast body of theoretical knowledge. We may not know which KK theory, if any, describes reality, but a large number of possibilities have been studied, and it's not obvious which paths one should mention.
The question refers to string theory, but string phenomenology, the application of string theory to the real world, does not employ extra dimensions in the way that old Kaluza-Klein theory did. As discussed in
this thread, the old KK theory obtains gauge fields from a kind of symmetry of the extra dimensions (continuous isometries) that the spaces now used in string phenomenology (e.g. Calabi-Yau manifolds) do not even possess. In the old KK theory, gravity is fundamental and the gauge fields actually result from gravity in the compact extra dimensions. In string phenomenology, one either starts with a big gauge symmetry that is just as fundamental as gravity (e.g. E8 x E8 in the most popular heterotic theory), which is then broken to something simpler by compactification; or one has gauge groups arising from the open strings attached to stacks of branes (as illustrated in comment #2 in this thread).
The old KK option still exists in string theory, and has been studied, but not for phenomenology, not for the purpose of directly describing the real world. Instead, it has been a way to learn more about the properties of the theory, how strings work. I know
@arivero to be interested in the possibility that the old-style KK mechanism could be applied within string phenomenology, and he has some interesting concrete ideas. So we may want to discuss that, along with, or even instead of, a review of how conventional string phenomenology works. After all, we do not at all know the limits of string theory, and we don't know that the way it is being applied now is on the right path.
We should probably also visit the field theory of electric and color charge, i.e. what "charge" means for the different cases of U(1) and SU(3). Mathematically it refers to a type of "representation" of the symmetry group. But where the representations of U(1) are indexed by real numbers - thus, the usual notion of electric charge - the representations of a group like SU(3) are instead vector spaces classified by dimension and by the action of the group on the vectors. So regarding the specific question
Spinnor said:
if electric charge can be roughly thought of as momentum in some compact space can the other charges be roughly thought of as momentum in a compact space in different directions from in our compact space?
I actually doubt that it can be thought of in that way. (Also, representing U(1) charge as a momentum in the fifth dimension has its own difficulties when applied to the real world, which I briefly mentioned in comment #2
here.) I think that, in an old-style KK theory of nonabelian gauge fields like SU(2) or SU(3), in the higher-dimensional description, the particles with weak charge or color charge would just be a many-component unified "spinor" field interacting with the gravitational field. However, there is actually a
classical description of such interactions, called Yang-Mills-Wong theory, that you might want to look up.