When Solovay showed that ZF + DC + "all sets of reals are Lebesgue measurable" is consistent (assuming ZFC + "there is an inaccessible cardinal" is consistent), there was an expectation among set-theorists that analysts (and others doing what you call realistic mathematics) would adopt ZF + DC + "all sets of reals are Lebesgue measurable" as their preferred foundational framework. There would be no more worries about "pathological" phenomena (like the Banach-Tarski paradox), no more tedious verification that some function is measurable in order to apply Fubini's theorem, and no more of various other headaches. But that expectation wasn't realized at all; analysts still work in ZFC. Why? I don't know, but I can imagine three reasons.
First, the axiom of choice is clearly true for the (nowadays) intended meaning of "set". Solovay's model consists of certain "definable" sets. Although there's considerable flexibility in this sort of definability (e.g., any countable sequence of ordinal numbers can be used as a parameter in such a definition), it's still not quite so natural as the general notion of "arbitrary set." So by adopting the new framework, people would be committing themselves to a limited notion of set, and that might well produce some discomfort.
Second, it's important that Solovay's theory, though it doesn't include the full axiom of choice, does include the axiom of dependent choice (DC). Much of (non-pathological) analysis relies on DC or at least on the (weaker) axiom of countable choice. (For example, countable additivity of Lebesgue measure is not provable in ZF alone.) So to work in Solovay's theory, one would have to keep in mind the distinction between "good" uses of choice (countable choice or DC) and "bad" uses (of the sort involved in the construction of Vitali sets or the Banach-Tarski paradox). The distinction is quite clear to set-theorists
but analysts might not want to get near such subtleties.
Third, in ZF + DC + "all sets of reals are Lebesgue measurable," one lacks some theorems that analysts like, for example Tychonoff's theorem (even for compact Hausdorff spaces, where it's weaker than full choice). I suspect (though I haven't actually studied this) that the particular uses of Tychonoff's theorem needed in "realistic mathematics" may well be provable in ZF + DC + "all sets of reals are Lebesgue measurable" (or even in just ZF + DC). But again, analysts may feel uncomfortable with the need to distinguish the "available" cases of Tychonoff's theorem from the more general cases.
The bottom line here seems to be that there's a reasonable way to do realistic mathematics without the axiom of choice, but adopting it would require some work, and people have generally not been willing to do that work.
Here is a negative answer to question 2 and the converse of question 3.
Let $T_1$ be the theory of the integers under successor $\langle\mathbb{Z},S\rangle$. This theory asserts that $S$ is bijective and has no cycles of any finite length. That theory is complete, since all models of size $\aleph_1$ are isomorphic, consisting of $\aleph_1$ many $\mathbb{Z}$-chains, and more generally there is only one model of uncountable size $\kappa$ for any uncountable $\kappa$. But there are countably infinite many countable models up to isomorphism, since every model consists of some countable number of $\mathbb{Z}$-chains. So $I_{T_1}$ is the function that says there are $\aleph_0$ many countably infinite models, but only one model each in any uncountable cardinality.
Let $T_2$ be the theory of infinitely many distinct constants $c_n$. There are countably infinitely many countable models of this theory, depending on the number of objects in the model that are not the interpretation of any $c_n$, but there is only one model up to isomorphism in any uncountable cardinality, since any model of size $\kappa$ has the distinct $c_n$'s and then $\kappa$ many additional points.
Thus, both of these theories have the same number of models in any cardinality: no finite models, $\aleph_0$ many countable models, one model in each uncountable cardinality. So $I_{T_1}=I_{T_2}$.
But $T_1$ is not interpretable in $T_2$, since if it were, the interpretation would involve only finitely many constants, but one can define only finitely many points in a model of the reduct of $T_2$ to those finitely many constants plus one parameter, whereas in any model of $T_1$, we can define infinitely many points relative to any parameter.
So this is a counterexample to the converse of question 3.
Note also that $T_1$ has no rigid models, since every $\mathbb{Z}$-chain admits translations, but $T_2$ has two rigid models, namely, the model with only the constants, and the model with the constants plus one more point. (Also, $T_2$ has models with only finitely many automorphisms.) These features show that $\text{Mod}(T_1)$ and $\text{Mod}(T_2)$ are not equivalent as categories, so this provides a negative instance to question 2.
Best Answer
"The complex numbers are easy to deal with, whereas for the integers it is much harder..."
What you might have heard about is that the complete first-order theory of the complex numbers with just the ring operations -- which is the same as the theory of algebraically closed fields of characteristic zero -- is decidable (i.e. "computable"), as was first proved by Tarski. So in principle one could write a computer program where you could input any first-order sentence in the language of rings, and in a finite amount of time it would tell you whether or not this sentence is true in the complex numbers.
However, if you look at the ring of integers, no such thing is true; the first-order theory of the integers is undecidable. This is a theorem of Alonzo Church, and is closely related to Goedel's famous incompleteness theorem.
The negative answer to Hilbert's Tenth Problem is a different issue -- this doesn't follow immediately from Church's Theorem, and was proved much later, by Davis, Putnam, Julia Robinson, and Matiyasevich.
I think of these things as more "logical folklore" than model theory, per se -- any reasonable introductory book on mathematical logic (e.g. Enderton's A Mathematical Introduction to Logic) will have a lot to say about them.