You really shouldn't ask so many questions at once, but to be fair many of them are connected. Indeed, almost everything I'm about to say can be summarised as follows: axioms are more or less useful in specific contexts which objectively exist in our experience, and trial and error gradually leads us to increasingly useful choices, which none of the concerns you mention really undermine. If they did, our civilization wouldn't have turned out the way it has. That's the tl;dr.
- Can new generations create different axiomatic models?
In both mathematics and empirical sciences, we choose axioms whose consequences are useful for what we're working on. Borrowing a term from Daniel C Dennett's description of natural selection, we might call the best options, the ones we're naturally drawn to over time, "good tricks". The world is a certain way, and sooner or later practitioners of the scientific method would, through trial and error, find more and more useful axioms to understand that nature. Were humans not so capable, we wouldn't have seen scientific advancement on so many fronts in our world. (It's no good accusing me of circularity in arguing induction works because it has so far, because my reasoning is abductive: the simplest explanation of past successes is that nature is comprehensible - not that we can articulate what an incomprehensible world would be like, of course.) But this includes certain mathematics. For example, you can't help but find symmetries, and hence group theory, relevant across physics, and in particular in special relativity - which electromagnetism motivates - a generalization of groups called gyrogroups is a good trick.
- If yes, can different axiomatic models create different models of (a) mathematics (b) science? If yes, is this a flaw of (a) mathematics (b) science?
On the one hand, yes, you can use different axioms. For (a), you can replace addition with Tsallis q-addition; for (b), you can pretend gravity obeys a different force-distance relationship. But we don't just coin axiomatic systems, we use them, and we find some pay off more than others in specific contexts. It's not a flaw; it's not a bug, it's a feature.
- Does the unprovability of certain things such as the unidirectional speed of light leave fundamental holes in (a) mathematics (b) science?
We could discuss specific examples endlessly, but the crux is the most testable models are the ones that make the most tractable assumptions to address present concerns. In this example, why would you assume light travels in different speeds from A to B and from B to A, and similarly from C to D and from D to C, and yet no matter which way you point your apparatus the averaged bidirectional speed is the same for everyone? There's no theory that makes sense of why nature would be like that. By contrast, an isotropic speed of light is easily motivated by a theory that classical electromagnetism calls for, and which is empirically successful in other ways. In science, we respond to problems with ideas, and test those ideas against their other effects; an idea without other effects, such as light conspirising against us to seem like its speed is isotropic when it isn't, isn't good for anything, so it won't gain acceptance under such criteria.
comment made by Ricky Gervais
Only true if we bother using the scientific method again fast enough, which an existing answer sociologically disputes, but that's not the point of your question. Gervais's real point is how this compares with, say, religions (each of which makes unique claims) and art/literature/music (which creates similarly unique works). There wouldn't be another Picasso without Picasso, but science is different because it uncovers how the world is - well, it probably approximately uncovers how the world is, before anyone gets pedantic and needlessly skeptical.
The choice of attributing convergence or divergence to a series is a purely aesthetic choice
Multiple definitions of the sum of an infinite series have interested mathematicians, but choosing one over another in a context is not about aesthetics. Mathematicians developed some theoretical insight into how these definitions can and can't work - in particular, which combinations of desirable properties can be combined, and which can't. Why are those ones desirable? Because they let us deduce the sum of one series from the sum of one or more others. But once all that's done, we can find certain contexts call for certain definitions. Physics loves one of them in particular, for this for example. The Casimir effect is measurably as strong as zeta function regularization lets us compute.
analytic continuation may be an example of there being no correct answer
A function can only be analytically continued in one way. I'm not sure whether your point is we might consider alternatives to analytic continuation, or might encounter functions with no analytic continuation. I think you already know my answer to the first; in case of the second, the functions without an AC tend not to unavoidably arise in situations where we need an AC. As a more general rule, functions that describe the world in scientific theories have the mathematical properties needed for the theory to be consistent and predictive, and the fact that many other functions don't have them isn't a problem; in fact, it helps us constrain what theory claims.
Despite these theorems, a lot can be proven in specific theories; in our world, they often have been. If a cultural reset didn't lead to as much success, that may signify our descendants' bad luck, but not a very constraining fudnamental limitation on mathematics or empirical science.
What do these limitations tell us about our reality if even the most sophisticated tool that is presently known to us (i.e. scientific reasoning) may be subjected to variations in definitions and may be characterised by an element of uncertainty?
The incompleteness theorems aren't deduced from our reality, so tell us nothing specifically about it. It is true other axioms would have different results, but your focus has primarily been on mathematics rather than empirical sciences, whose methods make it easy to sniff out increasingly useful choices of axioms. But even in mathematics, such experimenting works. For example, in set theory we restricted comprehension for consistency, and added replacement, slightly more controversially added choice, and have ever since preferred to classify large cardinal axioms by their consequences rather than adopt them as a matter of concensus.
If we compared our scientific books with the books of an alien civilisation, to what extent would they be the same?
How advanced are they? If we visit them, then find them at a Neolithic level, they won't have such books. If they visit us, they understand enough physics for travel across vast distsnces. In that case, we are right about a lot, and they a right about a heck of a lot, and there will be much overlap. They won't, for example, disbelieve in electrons.
Is mathematics invented or discovered?
At the risk of controversy, I'll defend one specific answer: neither. Unger and Smolin 2015 combine two dichotomies: rigid vs non-rigid properties, and prior existence vs none. In their terminology, we discover that which meets the first option in each dichotomy, and invent that which meets the second in each. But mathematics is a third option they call evoked, meaning it lacks prior existence but has rigid properties once adopted. They also consider a fourth option they call fictional, which has prior existence but no rigid properties, such as biological species: they clearly exist, but out attempts to define them have to be somewhat malleable.
In Chapter 6 of Philosophy's Future: The Problem of Philosophical Progress, Pigliucci cites this four-category approach as an exampe of how one can make progress in philosophy by advancing a debate with an expanded taxonomy of ideas. It's also interesting for our present purposes because it allows us to summarize my position herein even more concisely as "we can do very well by evoking axiomatic systems to taste". Unger and Smolin also mention chess is evoked, and that provides an interesting analogy for our present purposes. You can invent endless alternative games, but humans are bound to come up with fun ones, so how our psychology defines "fun", unless also radically changed, constrains which mathematically possibe games we'll play. Admittedly, this is nowhere near enough of a constraint to imply a very chess-like game will emerge (I'll leave it to sociologists, historians etc. to say how simillar the most popular board games have been in different times and places). But physical reality is much more particular than the wide and personal range of things we find fun, which is why we'd expect a lot of science to be "unique" in practice.