Now, can we imagine something which we would like to call "life" having what we would like to call a "metabolism" which use that sort of chemistry instead? Or is there some chemical reason why this is inconceivable?
I don't know if you can "make a living" on acid-base chemistry, because I don't think there's a high enough change in free energy to exploit.
For the most part - and I obviously have no way to be sure this is true, just as you have no way to be sure it's not - I think that the biochemistry of life on Earth, particularly of microbes, is so broad and diverse that it essentially encompasses the width and breadth of all the ways it's possible to "make a living" as a chemistry-based organism in this universe. If it's possible to make a living off of acid-base chemistry we should be able to find an organism on Earth that is.
Could our mitochondria be an example? ATP synthesis in mitochondria is driven by the energy stored as a substantial pH difference between the matrix and the intermembraneous space. The pH gradient is established by a series of proton pumps that are driven by energy from the TCA cycle (if I'm remembering this right.) So there's a way we're all making a living off of pH changes. But, say, the neutralization of acid by base? It's possible, but maybe we don't see it because the little guys get their lunch eaten by the redox guys who outcompete them.
Or is there some chemical reason why this is inconceivable?
I wouldn't say "inconceivable", but perhaps unlikely. Of course, the one thing that's always true in biology is that you can't say that anything is always true in biology.
My gut tells me that viruses should be categorized as life
As full-on life? My gut tells me they're life-like.
I don't expect you to ignore your gut for mine, of course, and I'm happy to accept all forms of disagreement on this issue. I'm certain that my definition will eventually be revealed to exclude something I "know" is life and include something I "know" is not. Yours, too. It's the problem with definitions.
What if this doesn't involve redox chemistry?
Well, I mentioned batteries because batteries operate by redox. (All forms of battery. If it's not redox, it's not a battery. Capacitors, for instance, are not batteries.) But, say, nuclear-powered self-replicating robots? I'm prepared to accept that as life even though we're now talking about "organisms" that engage in no chemistry whatsoever. But I think I'd create a different definition for such creatures.