One can posit mathematical string theories in any dimensions of any kind.
However, I do not understand why there are ten dimensions and not just any other number?
The specific dimensions arise from the requirements of the known physics encapsulated in the Standard Model and other data coming from particle physics, plus the requirement of General Relativity and its quantization. The Special Unitary groups whose representations accommodate the SM need at least these dimensions. There are models with more dimensions than this.
Also, if all the other dimensions are so coiled up in such a tiny space, how do we distinguish one dimension from the other?
We cannot move into the coiled ones, only in x.y,z. We do not need to distinguish them, as we do not distinguish the molecules in the air. The predictions from this type of theory on the behavior of particles is the only way of checking for their existence: consistency of theory with data.
If so, how do we define dimension?
A space variable ( centimeters) or time one ( seconds) that is continuous and maps the real numbers, each dimension at 90degrees to the rest, an extension of how we define normal x,y,z.That some are curled should not bother one. The coordinates over the earth are curled over the sphere's surface, for example, though the 90degree does not hold there. It would hold on the surface of a cylinder , z from -infinity to infinity, x from 0 to 2*r*pi.