@darij grinberg - This is the mistake in your initial reasoning: if $1\leq i\leq n$ and $\mu'\in M'$, then $c'_{i,\mu'}=(\mu'_i+1)\cdot c_{\mu'+e_i}$. This mistake - the missing nonzero coefficient - is immaterial, provided your argument, with the lonely diagonal product at the top of the lexicographic hierarchy of products in the determinant, holds water.

@darij grinberg - This is a promising way to attack the problem, though your reasoning at the start is incorrect. Note that the characteristic is 0 since the ground ring is the polynomial ring in the variables $c_\mu$, $\mu\in M$, with integer coefficients.

@Fred Rohrer - Yes, that's where I saw it! Thank you.

@Jeremy Rickard - Thanks. I googled "trivial extension algebra" and found some references which, however, all consider only special instances of the construction described in the question. Whatever, I got what I asked for: the construction is no longer nameles and I have a sensible notation for it. About the notation, borrowed from semidirect products of groups: this is more than just a distant analogy, see the continuation of my question.

@Fred Rohrer - Thank you, found it. Though this is not where I saw it before. I believe it was somewhere in Tsit Yuen Lam's "A First Course in Noncommutative rings", but for the life of me cannot locate it.

@Anthony Quas - Stupid me: meta.mathoverflow.net gets me there. I am hurrying home right now. Tomorrow I will cut the second part of the question out and repost it on the meta site. Thanks.

@Anthony Quas - I think so. I would rather post it there, it is inappropriate here. But when you are in pain, you don't care for appropriate manners, you just cry out... How do I get there, to meta.mathoverflow.net? Below on this page there is only the link to Meta Stack Exchange.