Hi MRAKGR, for your cube with 2 faces deleted,

re:
> As a way of simplifying the problem, maybe I could understand if you could explain how to Planar this
> cube which had two of its adjacent faces deleted.

For that case you would need to draw in a line and make one planar face by selecting the line and 3 edges using Construct > Planar, then join that, then for the final plane you can select the entire object and use Construct > Planar to fill that last one in.

But for the actual object when you do the boolean it leaves behind half of the planes that you need so you can use those and you don't need to draw anything extra since when they are joined the remaining planar holes are isolated from each other and so can be filled in with one use of Construct > Planar with the main object selected.

- Michael