(1) In Abaqus/CAE the stress at each integration point is given by the field variable $S$. Note that in Abaqus, like many or even most FE solvers, the stress computed is the Cauchy (true) stress. So, for a material with local material directions given by the Global basis: $\sigma_{x}=S_{11}$, $\sigma_{y}=S_{22}$, etc. You will need to make sure that stresses are being written to the ODB during the analysis to visualize them in CAE.
(2) This question is a bit more complicated. Generally, when talking about constitutive modeling the material parameters are independent of length scale. This is kinda the point of continuum scale modeling. If the material properties are varying spatially in your model you will need to either implement field variables or a user defined material (either a UMAT for Abaqus/Standard or a VUMAT for Abaqus/Explicit). I don't have any experience specifically with fiber reinforced composites in Abaqus, but the set of parameter you are describing in your question are for a general linear elastic, anisotropic material, which may or may not be able to describe the homogenized response of the material you are interested in. If that is the case, I know that it is possible to construct layered composites within Abaqus, but I can only point you to the Abaqus User Manual for more information. As for specific material constants, the best place to look is in the literature or, better yet, go run some experiments. This might not be the answer you are looking for, but each material is different. Given the anisotropy of the material you are going to want to look for data from multiple loading configurations.
I'm also not quite sure why you would need the volume for each element, but if your structure can be meshed uniformly them it might be possible to calculate element volumes prior to the analysis. If your mesh is irregular the answer is a bit more complicated. One possible solution I can think of is extracting integration point coordinates within the UMAT framework for each element, then based on all the integration points for the element determine the element nodal coordinates based on the associated quadrature rules. Another possibility is to look at the input file (.inp) and determine the nodal coordinates for each element, using them to calculate the element volumes.
(3) You can assign local material orientations in the Property menu of CAE. The symbol looks like a yellow L with red axes. Within the Material Orientation menu you can create local directions based on any coordinate system. In the Definition drop down select Coordinate system, then in the CSYS option create a new coordinate system. This can be down by clicking the coordinate system icon.
