I admit that more work would be needed to transfer all
results about

quantified modal logic (QML) to Dunn's semantics. But
there is a very

clean and widely used subset of QML that is trivial
to derive from

the laws and facts.

In particular, there is
no restriction on what language is used to state the laws and facts. For any world w, the facts F can contain any and every FOL proposition that is true about w, and the Laws L can contain any subset of F that is closed under deduction.

John, I'm just not following you here. Dunn's semantics is a semantics for propositional modal logic. There is just no purchase to the idea of sentences of first-order logic being "true" about a world. Moreover, laws and facts are not really even "stated" in Dunn's semantics. The law/fact pairs that replace the worlds of a Kripke model are not in fact sets of sentences but mappings from sentences (in the language of propositional modal logic) to truth values.

I admit that one might want to support conditions such
as Barcan's

formula for interchanging the order of modal operators
and quantifiers.

I haven't explored the details of how and whether
that work can be

transferred.

This is exactly the sort of thing that is determined by all of the apparatus that is missing from Dunn's semantics. Notably, the validity of the Barcan formula depends on whether one has a fixed domain of quantification for all worlds (or law/fact pairs) or world-relative domains whose membership can vary from world to world.

But before spending any time
worrying about such issues, I would like to see any evidence that they
serve any useful purpose in understanding the semantics of natural
languages as ordinary human beings (i.e., people who have not been exposed
to philosophers or logicians) use them.

Well, the central concern of this forum (I think) is not the semantics of natural language but knowledge representation. So I should think the relevant question here is whether worlds can play a useful role in representing information. Matthew's work, for example, suggests they can, as do certain of Lewis's own applications.

As far as counterpart
theory is concerned, my preference is to assume

(1) two individuals
in different worlds with the same name are likely

candidates for
being counterparts, and (2) if their significant life

events
correspond, they are the same, barring some exceptional

conditions that
are being discussed by whatever philosophers are

discussing them.

The appeal to counterparts to deal with *de re* modality is one of the reasons I myself find Lewis's account unacceptable, as I am not a modal reductionist. I should think you aren't either.

As for the question about substitutional interpretations of
quantifiers

and counterparts, I would like to note that the puzzles
and paradoxes

do not arise in finite or countable domains.

I'm not sure what this means. Are you saying that a paradox can only arise on the assumption that there are uncountably many things? That's isn't so. Russell's paradox can be generated inside a very weak set theory that does not include an axiom of infinity simply from assumption that there is a set of all sets. And semantic paradoxes like the Liar have nothing to do with cardinality whatsoever.