Blender how to remove extra edges when cutting out a shape in a mesh?

Question

Blender

1. Cut out the shape of the cube for the window from the mesh using the "logical" operation.
These edges appeared
2. Because of which the "subdivide" operation does not work correctly

3. It is impossible to dissolve the edges, the following situation occurs when deleting the edges

4. Even if you bring to mind and under dance with a tambourine to remove these faces, then the new solid face is not subdivided
. Questions:
What is the best way to create holes in meshes to avoid these problems?
And how to fix this problem with the ribs?