I'm not sure what might be considered a canonical reference, but see for instance Section 1.7 of M-Completeness Is Seldom Monadic Over Graphs, which describes foundational issues regarding treating $\mathbf{CAT}$ as a category versus a 2-category.

If you're planning to write a paper, there's no advantage in trying to write out the pronunciation: you may just use the symbols. If I was trying to pronounce the name, I would say "tensor-not category" or similar, analogously to "dagger category" for †-category or "star algebra" for *-algebra.

It's unclear to me whether or not this is true: the definitions are very abstract, so it's difficult to get an intuition without explicitly working through the example. I notice that Bourke & Garner do not seem to give any examples in this vein, but it's not clear whether this is a deliberate omission. I personally would be interested to know the outcome if you do work through it.

Interesting. Your theories are a little unusual, as they no longer look like a class of categories equipped with specified structure and generators, as in the traditional setting. If you haven't checked already, it would be worth comparing your setting to that of Bourke & Garner's, which establishes theory–monad correspondences in a great level of generality (potentially at the cost of some massaging to check your definitions align with theirs).

Could it be the case that the equivalence works out in the $\mathrm{Set}_*$-enriched setting, giving a correspondence between $\mathrm{Set}_*$-enriched cartesian operads (or substitution monoids) and finitary $\mathrm{Set}_*$-monads on $\mathrm{Set}_*$? It looks like your setting is one in which sets are being consistently replaced with pointed sets, which seems suggestive of the setting for enriched Lawvere theories. But perhaps there's an obvious reason this doesn't work out.

One last reference: the introduction to Cockett–Santocanale's On the word problem for ΣΠ-categories, and the properties of two-way communication is a nice survey, and also describes this category in terms of games. As far as I can tell, there hasn't been progress on this since, which would suggest the problem is still open.

Hughes' A canonical graphical syntax for non-empty finite products and sums treats nonempty finite products and coproducts on discrete categories, where they state that (at point of publication) a categorical formulation of the result for non-discrete categories and empty products/coproducts was an open question.

A syntactic construction of the free finite product/coproduct bicompletion is given in Cockett–Seely's Finite sum–product logic. There are also the earlier, suggestively-named papers Free bicomplete categories and Free bicompletion of enriched categories of Joyal, though I couldn't find these online. Hu–Joyal's Coherence Completions of Categories and Their Enriched Softness describes the free completion under products, coproducts and a zero object.

Using small presheaves instead of arbitrary presheaves, as Shulman does, is another way to avoid the size issues, allowing you to avoid considering relative pseudomonads. (Though this seems a little less elegant than the relative setting to me.)