You’re welcome!
1) I’m not exactly sure what your question is here. You want to see which modes are supported in an optical fiber? The modes at the output of the fiber actually only consist of the fiber modes. The input mode spatial data is used to determine the different power levels (coupling coefficients) attributed to each fiber mode. If you are interested in solving for the different modes of a radially symmetric fiber, I would really suggest looking at OptiFiber. It is an extremely easy tool to use and will calculate the fiber modes, as well as give pertinent information. It can also be used with OptiSystem.
2) The spatial distribution of the fundamental mode (or any mode) is a complicated problem, which can sometimes be solved analytically. It depends on the refractive index distribution of your waveguide and is usually calculated by a mode solver. But yes, generally the spot size is determined (if the fiber is a step-index fiber) by the core radius.
3) Exactly!
Hope this helps!