Quasigroups, congruences and recognizable subsets
(1) for any function $\varphi:E\to F$, the relation $R_\varphi$ as you define it is always an equivalence relation on elements of $E$. But the notion of congruence depends of the laws you have on your sets, and $R_\varphi$ will be a congruence for the operations that are preserved by $\varphi$. Notice that your formulation with $usv$ is not anymore well-defined without associativity: you have to state it separately for $us$ and $sv$.

(2) Let us weaken even more the hypotheseses. Let $\varphi:E\to F$ be any function, without any structure assumed on $E$ and $F$.
Then if we take the definitions you wrote for "recognised" and "saturated", the 3 propositions are still equivalent, as your proofs still work.

The thing you will lose by considering arbitrary sets (or groupoids or loops) is that you won't carry out the nice monoid structure of $\Sigma^*$, so you lose almost all the tools of regular languages theory, like Myhill-Nerode equivalence, Green relations, and so on...

For (3), I think Jean-Eric Pin's webpage (including the document you mention) contains good surveys of this field.