String Theorys Dimensions
As far as I know (and again, I am only beginning to learn string theory) the dimensions exist much like any other dimension. The particle can "move" in the dimension, just like it can move in 3 dimensions + time, so in that sense they are physical. However, they cannot be observed (as they are compact) so, we are basically living in a 3+1 dimesnional world: the other dimensions are "stitched" to it, but are, well, somewhat different. You can't realy perceive them, but they have physical affects (such as determining the electric charge) so they are important.
In mathematical terms, the space-time is composed of a flat 3,1 space, multiplied by a conformal two-dimensional space with c=22 (c being the centeral constant).
In human words, you can say the world has 3 dimensions + time, but every point in space also has far more many "internal" degrees of freedom.
I think.
Regarding the difficulties of the theory, (super)string theory is basically designed as an approximation of a true theory, and the difficulty is ascertaining what is the exact (i.e. not approximated) theory. This theory (that no one knows) is called M-theory.
According to certain vaccum expectation values (i.e. measurments of the most basic entities that exist in the world) the exact string-theory can (presumely) be found that describes our world. But without M-theory to give us these values, we are not able to find the exact values, the exact string theory, that will describe our world as we know it. So one problem is choosing the right expectation values - ones that will result in a valid string theory. As there are infinitely many string theories, and it is misleading to interpert them using naive approaches, this is not at all easy. Another problem is to find the M-theory - which will make the former problem trivial. Good luck on that, no one has a clue how IT even looks like.
