2. What is a Solid?
As far I've was able to understand it is another way to define a 3D object (vertices, faces, color(materials)).
It was defined by D. H. Laidlaw, W. B. Trumbore, and J. F. Hughes. "Constructive Solid Geometry for Polyhedral Objects", 1986.
Commonly known as CSG.

Besides the CSG method there is another one: Boundary Representation (B-Rep). That is what is used in Sweet Home 3D, or rather used through the .obj definition.
B-Rep defines a model by specifying the limits of its volume: a collection of connected faces and their edges and vertices.

There is a free opensource program called OpenSCAD that you can use to import an .obj file and export is as CSG. I'm not sure what you want to do with the resulting CSG because it's useless in Sweet Home 3D.
Check out FreeCAD (opensource). You can import .obj files and it can execute several of the operations you want like intersection etc.

4. What I looking for?
A way (simple, eventually), to get the current vertices values of a selected piece of furniture to make a CSGObject.

To simply get the vertice values: each v line in an exported obj file represents a vertice of the 3D model. I haven't checked but I'm pretty sure the PieceOfFurniture object in Sweet Home 3D holds exactly that obj definition in some form, maybe twice: original and transformed.