You seem to be tackling several issues at once. First though, some inaccuracies. You write "when creating a system of axioms like these..." I'm not sure what 'these' refers to. Then you say "it's necessary the list of axioms is complete." Do you mean by 'complete' that there is only one model of the axioms (up to isomorphism)? if so, why is that necessary for modelling probability events? You comparison with the axioms of geometry is unclear as well. If you omit the fifth, you do not automatically get hyperbolic geometry, you can also get projective geometry. To claim that any of those is not what we wanted to have is peculiar, particularly from a modern perspective. Geometry encompasses much more than just Euclidean geometry. And again, even with the fifth there is not just one (up to isomorphism) Euclidean geometry, but infinitely many (of various dimensions).
Now I will try to address the question of what is so great about Kolmogorov's axiomatisation. The mathematics of probability is fraught with difficulties, both conceptual and technical. There are endless examples of seemingly simple questions that turn out to be very complicated or have severely counter intuitive answers (The Monty Hall paradox for instance). Problems that appear identical may turn out to be significantly different just because of changes in the protocol. In short, it's not easy.
Having said that, the probability theory of finite probability spaces is quite simple, at least in the sense that it is clear how to model finite probability spaces: Given a finite set of events, the probability of a subset of events is the ratio of that subset to the entire set. Sweet. From it flows quite a lot, but only when the total set of events is finite.
Often, the set of events is infinite. For instance, modelling throwing a dart at a dartboard is often done by imagining the dart board as a disk in $\mathbb R^2$, and then a throw of a dart corresponds to a choice of a point in the disk. Of course the disk has infinitely many points. What is the probability that the dart hits a given point, say the centre of the disk? Well, assuming the dart lands randomly at a uniform distribution over all points, the only possible answer is $0$. A point is just too small. This is already counter intuitive enough and raises the question of how to model all of this. Well, this is all related to the notion of how big a set is. An innocent question with a highly complicated answer. It's not simple at all to develop the theory that answers this question - measure theory. Issues related to the axiom of choice quickly creep up. A famous theorem of Vitali shows that it is impossible (assuming the axiom of choice) to meaningfully assign a measure to each and every subset of $\mathbb R$.
Now, measure theory was not developed to provide some foundations of probability theory. Instead it arose from questions of integrability. Kolmogorov's wonderful insight was that he realised the same formalism can be used to turn the intuition of what probability theory should be (as you say, pretty obvious axioms) into actual axioms. Before measure theory and Kolmogorov's seminal contribution nobody knew how to meaningfully and accurately work with infinite probability spaces. Thanks to Kolmogorov a formalism was born. Now that is truly wonderful.
Lastly, the paragraph you quote is talking about something all together different. Quantum mechanical considerations defy many conceptually obvious properties. Among them Kolmogorov's axiomatisation of probability. In the world of quantum mechanics even probability behaves differently than what we are used to. Such is life.
It's slightly orthogonal to your question, but I would consider to take a traditional course in ordinary differential equations as the "next thing", rather than assuming a theoretical calc course (whether very theoretical [real analysis] or medium theoretical [advanced calculus]) is your natural next step.
DiffyQs has way more utility since it is needed in engineering and physics, as well as math. Whereas, you can get by, certainly at undergrad level without real analysis. And even most grad engineers and physicists don't need RA either (except for extremely math inclined theorists). So do the ODE first. If you liked integration and the tricks and techniques there, you will enjoy the tricks and techniques of diffyQs.
For that matter, I would even be inclined to take an engineering slanted PDE course and engineering slanted complex analysis course before diving into RA.
Also, if you have not done so, consider to take a linear algebra course. Can be a pretty easy one ("matrices") rather than the most difficult. But get something. (This is nothing against the harder versions or the ones with more content, but just those can be taken later.)
Best Answer
As a direct answer, calculus has no axioms inherent to itself. Theorems of calculus derive from the axioms of the real, rational, integer, and natural number systems, as well as set theory.
Most disciplines of modern mathematics exhibit this sort of behavior, in which the discipline has no axioms inherent to itself. Modern disciplines of mathematics typically work under a unified axiomatic system, the most common one being Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). ZFC is powerful enough to encode our most frequently encountered structures, including our various number systems.
What, then, do the various disciplines of mathematics working under the axioms of ZFC study? The answer is the various structures that can be constructed using the axioms of ZFC. For example, single-variable calculus can be (very) broadly characterized as the study of real-valued functions with real domain. Such functions can be defined under ZFC, because the real number system can be axiomatized in ZFC.
Similarly, group theorists study algebraic structures called groups, which are defined in ZFC as a set (the building blocks of Zermelo-Fraenkel set theory) and a binary operation on the set, which is a function (which is also defined in ZFC using only sets) obeying certain properties. In a sense, one could interpret these properties as axioms for group theory, but they are actually merely definitions - the underlying axioms are those of ZFC.
The question of truth is a bit trickier to grasp, and there are many differing viewpoints. Complicating the question is that one could adopt an axiomatic system other than ZFC. If you take the perspective that true propositions are those that can be proved in your favorite axiomatic system, then it could easily be that a mathematician that believes in ZFC will disagree on the truth of a proposition with a mathematician that adopts a different system. And different axiomatic systems are not strange or unnatural - indeed, they constitute one of the major objects of study in mathematical logic.
One might also adopt the viewpoint that axioms are not self-evident truths, and therefore one should not believe or disbelieve in any particular axiomatic system. For many of these people, a collection of axioms is a list of meaningless rules to follow, and mathematics is the game of manipulation of symbols under these meaningless rules. (This is Hilbert's formalist perspective.) Truth, then, is a meaningless concept - all propositions are outcomes of some game with some particular rules.
There are other perspectives on truth, many of which I am not familiar with. But this should convince you that the concept of "mathematical truth" is more nuanced than you are probably aware, and really varies from person to person.
As for my personal opinion: I lean toward formalism, so to me it is meaningless to talk about whether calculus "gives me truth." Being inclined toward analysis and topology, I view calculus as an incredibly important and useful tool that agrees with my intuitive understanding of the world, but I view the underlying axioms as just a list of rules I am allowed to play with, without regard to whether I believe in their "truth."