First, there is nothing unusual about having a theory with infinitely many axioms. This is the case for both of the two most commonly studied "foundational" theories, Peano Arithmetic and Zermelo-Fraenkel set theory. In both of these cases, what is often described as one named axiom (induction for PA, selection and replacement in ZF) are actually axiom schemas -- that is, each is just a recipe for generating an infinity of different axioms by plugging different logical formulas into the schema.
First-and-a-half, remember that the Gödel procedure for producing an unprovable (and un-dis-provable) statement requires not only that the axioms is effectively generated, but also that the theory is rich enough to be "interesting". (For example, the theory of integers and addition where multiplication is not mentioned is not rich enough for Gödel's construction to work on it).
Second, if your original theory $A_0$ is effectively generated, then your $A_\infty$ will be too -- there's a mechanical, deterministic process that will eventually print every axiom in $A_\infty$.
$A_\infty$ will be consistent too, if $A_0$ is -- by the "compactness" property of formal logic which says that every inconsistent system of axioms has a finite subsystem that's also inconsistent. (If there's a proof of a contradiction in the system, then because proofs are finite by definition, the proof depends on only finitely many of the axioms). Every finite subset of $A_\infty$ will be a subset of one of the $A_n$'s, and all of these are consistent by construction.
Since $A_\infty$ is effectively generated and (because it extends $A_0$) sufficiently rich, we can repeat the Gödel process on it, and get at Gödel sentence for $A_\infty$. Add that to $A_\infty$ to get $A_{\infty+1}$, and proceed ad nauseam. (In this context it is traditional to write $\omega$ instead of $\infty$, and there's a theory of the necessary numbers "beyond infinity" under the name "ordinal numbers").
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."
Best Answer
Suppose that a system of axioms is consistent. An axiom is compatible with this system if after adding it, the resulting system is still consistent. Yes, you should check whether adding this additional axiom preserves consistency. The technical name is known as equi-consistency. We say that the larger axiom system is equi-consistent with the smaller axiom system if the existence of a model of the smaller system implies the existence of a model for the larger system.
For the Axiom of Completeness, if the base axiom system is that of an ordered field, then we start with a model of the rational numbers and then construct a model of the real numbers using either Dedekind cuts or Cauchy sequences.
If you add an axiom to a base system and the resulting larger system is inconsistent, it means that the negation of the axiom is provable from the base system.