Your questions relate to the fundamentals of the Finite Element Method. I'll give a brief overview here of what generally happens and then discuss your specific case.
Typically, each element type (line, shell, solid) is associated with a specific mathematical formulation referred to as an isoparametric formulation. Basically what's happening behind the scenes is that each element type is transformed into it's isoparametric type, regardless of its geometry. (if you want to visualize this, one example is that a parallelogram shell element gets transformed into a perfect rectangle in this formulation).
During this process it also creates new specific points where your resultants (forces and moments for line types, force per unit for shells, stresses for solids) can be accurately calculated. It's worth noting that these resultants are NOT exact anywhere else on the element other than at these points (theoretically anyways. Practically we can interpolate them with a high degree of accuracy).
Here's the crux of your issue - there will be as many of these points (referred to as Gauss points) as there are nodes in an element. So a 4-node quad shell will have 4 gauss points, an 8 node solid will have 8 points. Now most software will translate the gauss point results over to the actual node positions (there are reasons for this but they're not very relevant to this discussion).
So there you have it. SAP2000 is showing you stresses at nodes because FEA only allows it to accurately calculate results at the gauss points and then interpolate them over to the nodes.
With regard to why solids only have 3 dofs at each node instead of 6, I'll refer you to Daryl Logan's excellent book on 'First course in FEA'. The simple reason is - the math checks out.