We can achieve this procedurally in the shader editor using displacement in only a few nodes.
We'll start with this approximation of your model.
In the shading workspace, we can add a texture coordinate and take the Generated coordinate space into a Separate XYZ node to look at our object over different axes over a 0-1 range. In this picture I'm looking at ...
Let's say that you want to sculpt these two objects together:
First, apply their scale, then apply all their modifiers except Subdivision Surface that you just disable. Keep in mind that when you join objects, the last selected while keep its modifiers (and they will affect the joined meshes) while the others won't:
Join them with CtrlJ, then select all ...
Boolean is a solution, but you could select, duplicate and separate a part of the first object (here the topology is messy because I've sculpted it with Dyntopo but it would be the exact same process with a clean mesh):
then extrude and sculpt (or model). Make sure that you don't alter too much the bottom so that it continues to fit well:
Don't apply the subdivision modifier. This makes the geometry unnecessarily dense. Re-meshing may have solved your problem but this will not hold up.
Also smart UV unwrap isn't the best way to unwrap because of lack of control. Instead make creases and then unwrap using U>Unwrap>Unwrap. Also learn about edge-flow. This is very important when modelling.
Simply add a Solidify Modifier and add thickness to the cube. Do bear in mind, this does essentially double the geometry calculated at render, so this might increase render time some.
This is the result from your file:
In the screen shots you sent it shows you in "Object Mode", you can't paint or sculpt in this mode.
To paint textures you have to go to "Texture Paint", and have the proper setup to paint.
To sculpt you have to go into "Sculpt mode", and a object you can sculpt with