I am still unable to find a reference for this. Have you been able to find an answer to this since?

Dear @Z.M, I am actually starting with $B$, and I am mostly interested in the transcendence degree of its residue field, which I think is only really linked to the 'classical' Krull dimension. I just realized that for the cases that I care about the most, the completion happens to be affinoid. I phrased the question as above because it becomes a standalone question which experts might be able to answer.

Dear @Olivier I can mostly see this (although I would still need to thoroughly check the proofs, especially the non-commutative case), but I am not confident enough to write this in any official mathematical text. That is why I am asking for a reference.

Also, why is the infima of seminorms still a seminorm? I thought that one would lose the subadditivity.

Thank you for the comment, @TimCampion. I was not aware of the result about power-multiplicative seminorms. Does the "powermultiplicativified" norm still induce the same topology?

Thank you so much for the answer Martin! In the third question I totally forgot the initial object (as a "trivial case").