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try adding a \strut at either end of the text
Thanks for \strut ! It solved the vertical misalignment problem (although I haven't figured out exactly how ).

Using the 'calc' package also simplified the alignment of left and right borders so it would precisely fit \textwidth.

There is still ...

Greetings!

I can't seem to get the formatting right.
I want the side boarders of the inner boxes exactly overlap with the
borders of the outer box and still overlap in the middle as they do
now.
I also want the upper border of the box containing "aaa aaa..." to
exactly overlap with the surrounding ...