So, is there a way that I can get a view of which edges they are so I can adjust the input geometry?
Unfortunately, we don't have a very good way of visualizing bad topology. People tend to use “Divide” sop's “Remove Shared Edges” to inspect boundary edges that may look like interior ones (not applicable in this case). For non-manifold edges (those incident to 3 or more polygons), your best bet may be to use PolyDoctor's (disable everything in the first tab right away) “Visualize Maximal Manifolds” under the “Topology” tab. This will colour your polygons. The bad edges are where two different colours meet. You can also visualize inconsistently wound polygons by unchecking “Ignore Winding of Input Polygons” in the same tab. Your geometry doesn't have this issue would also be something quite detrimental to passing a geometry as a solid. If your input is a connected manifold you should get a single colour for the whole thing. The problem is of course that you may fail to see very small things, as is the case with your geometry. I spotted two problem areas (see the screenshots). If you enable “Create Manifold Number Attribute” and check the geometry spreadsheet you can see that your geometry has has 12 maximal manifolds, numbered 0 to 11. You need to have one, or generally, as many as the number of connected components of your geometry. If I wanted to guess, I'd say you have used the Fuse SOP
somewhere higher in the chain?
As for the rest of your explanation, I'm assuming that I need to define B as a surface instead of a solid?
Yes, if you are cutting A with B, generating only pieces of A, then you'd always be better off marking B as surface.