That is, are there certain conceptual primitives, such as object, action, structure, property, logic, event, quantity, partial, paradox, system, concept, etc, or connectives/judgements, such as for all, in which, or, false, there exists, such that, true, and, etc that rather than being merely tools to discuss concrete matters of logic and mathematics are themselves invariant and as such matters of study in their own right, and admissible to a concrete semantic model (or set thereof)?
The question in the title asks about a language, but the body of the question asks about semantic concepts, which are not the same thing, so I don't know what you're really asking about. So I will assume that you're asking about relations between natural languages (including hypothetical ones) and semantic concepts. "Paradox", for example, is not a primitive, it is a higher-level concept derived from "proposition", "truth", "denial" and "existence" among other things. Being "primitive" is different from "being an extremely convenient concept".
Some of these concepts, for example "object" (by which I assume you mean "thing") and "action" are indeed essential to human cognition, and a model of the mind or of a language which did not have analogous concepts would not be tenable. Some of your proposals are probably not universal, for example "system" is a rather advanced abstraction that is unlikely to exist in some languages. There probably is no word that replicates "system" in the Taa languages. That doesn't mean that speakers are incapable of grasping such a concept including taking the word "system" into their language as a way of expressing the concept, it simply means that there is probably no such pre-existing concept in the language. So then it could be of some interest to theories of epistemology, cognition and language to identify any truly universally-present concepts vs. those that might be missing in some language.
All connectives express some kind of concept, and are typically logical-jargon ways of succinctly verbalizing notations that are often used in logical formalism. They generally are not used as such in natural language. I would say that separating judgments from concepts is a mistake: judgments like "all", "true" are not really different from "tall", "fish".
Sure, there could be. There is. Predicate logic is a logical language, and it is universal among those humans who know it, inconsequential notational variations aside. I hesitate to say for sure whether it is the same for the entities on Arcturus IV, but it is hard for me to imagine that it will prove to be any different, out there.