Here, on page 4,
the phrase "the same ≤ r-element suboperations up to isomorphism"
means s-element suboperations for some fixed s≤r?
Is this interpretation correct?
Moreover, how can I see that
The strongest of these is ≤r-isomorphism, and the weakest of these is operational r-isomorphism.
The most obvious reading is that it is ${\bf every}\ s \leq r$, and this seems to be borne out by the definition of $\leq \! \! r$-${\it element\ function}$ in Def 1.2 and that of ${\it suboperation}$ in the preceding paragraph. Cf. ${\it operation}$ in Def 1.1.
Now, maybe this already answers your second question, but anyway, here is a way to break down the problem: There are four obvious pairs that you could compare separately, for example $\leq \!\! r$-isomorphism vs $r$-isomorphism, and $r$-isomorphism vs operational $r$-isomorphism. I would expect to find that 'not operational' is stronger than 'operational,' both for $r$ and for $\leq$, and that $\leq\!\! r$ is stronger than $r$, both for 'operational' and for 'non-operational'.