For your first question, see this discussion by Todd Trimble.

The section titles are proposed solutions, as I wrote in my answer. Consensus on synonyms is driven by usage. I'm not aware of alternative proposals for alternative terminology, and the terminology proposed on the nLab page seems reasonable.

Categories of $S$-sorted algebraic theories are defined in many references (Rezk is an unusual reference: the original reference is Bénabou). But Rezk does not appear to define a category of multisorted algebraic theories using the Grothendieck construction, so I do not see that this answers the question.

It appears the proofs of those statements are omitted from the paper entirely...

I don't know of any good references for either category of multisorted algebraic theories. People tend to study those of a fixed sort. You could try looking at Tarlecki–Burstall–Goguen's Some fundamental algebraic tools for the semantics of computation: Part 3. indexed categories, which studies some related constructions.

One correction: a multisorted algebraic theory is not a functor from $(\mathrm{FinSet}^S)^{\mathrm{op}}$. You need to restrict to the indexed sets with finite support: otherwise you may have infinite products when $S$ is nonfinite.

This is certainly one reasonable category of multisorted algebraic theories. The other has the function between sets and the product-preserving functor going in opposite directions. However, it doesn't make sense to ask which is "correct": they're both interesting to study.

*Brian Day, Elango Panchadcharam, and Ross Street.

@PaulTaylor: it is not necessary to assume that the underlying endofunctor preserves reflexive coequalisers, but when this is true (and $C$ is cocomplete), then in particular $C^T$ has reflexive coequalisers and so is cocomplete.

Yes, this is true.