The connected faces to edges of interest lie in the same plane. The edge angle is the angle between the normals of the two.
Can select all edges of interest by selecting one edge in edge selection mode, then in menu Select > Select Similar > Face Angle
then X Dissolve edges.
This is same behaviour as limited dissolve delimiting by ...
This is happening because you have some faces pointing inwards and some pointing outwards. You can tell by turning on Face Orientation and Face Normals in Overlays:
To fix it, select the whole thing in edit mode and then select Mesh > Normals > Recalculate Outside. You can now bevel as you wish:
From what I can glean from here, Blender's representation of a polygonal mesh does not currently permit faces to have holes.
The rule appears to be that a face must be bounded by a single loop of vertices, with no vertex visited more than once, so..
.. a hole can be bounded by two faces, as on the left, but not by one, as on the right. That would mean ...
If the plane and cube belong to the same mesh, in Edit mode, you can use CtrlF Face Menu > Intersect (Knife). It has 3 modes:
(Illustrations exploded to show separate parts)
All: all the results of intersection are split to separate islands
Cut: Edges are created, but not split, at intersections in all parts
Merge: Edges are created at intersections, ...
You work with too many polygons and you should think another way. If we try something like your last picture:
Create a UV sphere, stretch it on the Z axis, keep only 1/32 that you mirror on the X and Z axis:
Cut the hole with a knife:
Use an Array in Object Offset mode to make the mesh rotate around an empty, work on the mesh as long as ...