I read the articles in the archives about Kociemba's algorithm about
a year ago, without (I confess) fully understanding them. In particular,
I do not fully understand what differentiates Kociemba's algorithm from
Thistlethwaite's algorithm, other than it uses a different arrangement
of nested subgroups.

The basis is similar (although Kociemba's algorithm uses searching to
get solutions while Thistlethwaite's uses tables and directly arrives
at solutions). The main difference is that once a solution is found
Thistlethwaite's algorithm stops. Kociemba's algorithm continues finding
newer solutions (even longer than the original solution) to phase 1 and
trying to fit them with a solution for phase 2 such that the total solution
is shorter. This proves to be very effective. Of course this is easier to
do with a 2 phase algorithm than with a 4 phase algorithm.

But in the meantime, I wonder if you could verify that Kociemba's
algorithm does not guarantee to find a minimal process? In particular,
is it the case that 26q is a minimal superflip, or is it only an
upper bound?

Given time Kociemba's algorithm will find a minimal solution. I confess
that my implementations does not if the configuration can be solved
through phase 2 only, but the cube can be rotated to avoid that. The
reason is that ultimately Kociemba's algorithm will find longer part
solutions of phase 1 and ultimately stumble on a complete solution in
phase 1 which will be minimal because of the breadth first search.

But it can take long. Getting a minimal solution if the length is 16
or less appears to be doable. If the length is 19 or more it takes an
awfully long time. What I have found until now is:
1. There are no configurations known that require 21 turns or more,
and I checked an awfully large number.
2. There are known configurations that require 18 turns.
The middle part is a grey area.