No and no, I am afraid. Those things are not specific to probability, we do them with logical connectives, which parallel set operations, and with areas and volumes, just as well as with probabilities. And countable additivity is not necessarily natural for some classical applications (e.g. to natural number frequencies), it is a technical assumption. Indeed, Kolmogorov's Foundations of the theory of probability (1933), that axiomatized probability in terms of measure theory, does not ask that the collection of sets be a $\sigma$-field in the initial five axioms, only a field, and that the probability measure be $\sigma$-additive, only finitely additive. Countable fields only appear in chapter II, and he formulates the last axiom in terms of measure continuity, not additivity.
The concept of set field he borrows from Hausdorff's Grundzüge der Mengenlehre (1914, English translation), which is not addressing probability at all. And the idea of considering collections of sets (or rather classes back then) closed under standard operations goes back to Boole and de Morgan, who were not even doing measure theory, they were interested in logic. Measure theory was originally developed for the case of real number line, with Peano-Jordan, Borel and Lebesgue introducing set functions on systems of sets, whose properties were eventually packaged into set fields, $\sigma$-fields, and $\sigma$-algebras.
Borel was the first to require $\sigma$-additivity in Leçons sur la théorie des fonctions (1898), but his interest in probability only develops around 1905. Lebesgue took a more abstract view of sets beyond subsets of a line in
Leçons sur l'intégration et la recherche des fonctions primitives (1904), although he states his conditions in terms of integrals rather than measures. Neither Lebesgue nor Hausdorff were much involved with probability theory. A short review of this gradual measure-theoretic abstraction process is Le processus d'abstraction dans le développement des premières théories de la mesure by Villeneuve (it is in French but Google produces a decent translation).
For longer treatments see Classical and Modern Integration Theories by Pesin, Lebesgue's Theory of Integration: Its Origins and Development by Hawkins, and on connections to probability specifically Sources of Kolmogorov's Grundbegriffe by Shafer and Vovk, and Feller’s Early Work on Measure Theory and Mathematical Foundations of Probability by Fisher.