Your first question is handled in Freyd’s paper, which shows that the notion of “a binary relation controlled by a set” is richer than you would think. Freyd says a relation $\square$ from $L$ to $R$ between large sets is resolvable if there is a small subset $L’\subseteq L$ such that every $l\in L$ has the same $\square$-relatives as some $l’\in L’$.
Remarkably, resolvability of a relation implies that of its opposite! This is the key elementary foundation for everything in this paper. The proof is easier than thinking of the statement: for each subset $A$ of $L’$ for which this is possible, pick $r_A$ such that $A$ is the set of all $r_A$’s $\square$-relatives in $L’$. Then it’s easy to check that the (small!) set of all $r_A$s resolves $\square^{\mathrm{op}}$.
The categorical application of this notion is to say that a span $A\leftarrow X\to B$ $\square$-relates to a cospan $A\to Y\leftarrow B$ if and only if the square commutes. Isbell proved that a concrete category has $\square$ resolvable for every $A,B$, and Freyd proved the converse. This is saying that a category is concrete if and only if it doesn’t have too many distinguishable maps into or equivalently out of any pair of objects, which is a hyper-general (co)wellpoweredness condition.
If now the category has binary products, respectively coproducts, then it is equivalent to look at the analogous relations involving forks, respectively coforks, centered at a single object $A$, and one proves this directly. It’s actually the first lemma toward proving the concreteness theorem. Resolvability of these relations is what Freyd calls generalized regular (co)-well-poweredness, so we do know that these are equivalent for categories with binary products and coproducts; adding equalizers and coequalizers gets you the result you mentioned, and again it’s easy to prove directly that resolvability of the generalized regular subobject relation is equivalent to regular wellpoweredness when there are equalizers. So that handles your (1).
Toward your (2), I’m pretty convinced you can show these results are optimal by fiddling with Isbell’s original non-concrete category, which is a union of a large number of commutative squares spanning and cospanning two fixed objects $A,B$. This is (generalized) regular well- and co-wellpowered because there aren’t any nontrivial forks or coforks. Now freely adjoin a product $A\times B$. You haven’t added any coforks but $A\times B$ is at the middle of a large number of distinguishable forks, so this is a guy which is not generalized regular well powered but is (generalized) regular cowellpowered. Now freely adjoin equalizers and then coequalizers. Since we’ve never created any new coforks and every nontrivial fork is centered on $A\times B$, I claim that all this adds is an initial object and a terminal object plus a large family of distinct regular subobjects of $A\times B$. Now this has equalizers and coequalizers, is not regular well powered but is regular well copowered.
So I think the moral is that forks are less meaningful than squares in the absence of products.
