Is there any easy way to get rid of the visible triangles in the final rendering? It seems the subdivison in some places is more intense than in others. Or is the only solution to rebuild the mesh?
You will need to create loop-cuts from the corners of the doorway horizontally and vertically, so your shape will be essentially 6 'faces' but with the removed doorway.
This SHOULD fix your problem.
One thing you could do is research 'topology' and understand how shapes and vertices work together.