The sentence $1=0$ is false, so it is (hopefully) unprovable. On the other hand it is disprovable (in any halfway reasonable theory for reasoning about the integers) and therefore decidable.
In order to be undecidable, both the sentence and its negation must be unprovable. In other words, your "a sentence which is unprovable, and also its negation is unprovable" is exactly what "undecidable" means.
Note that this is always relative to a particular theory or proof system. Something can't just be "undecidable" in and of itself; but it can be "undecidable in ZF" or "undecidable in PA", for example
(Beware that in the related area of computability theory, "undecidable" has a completely different meaning, and is synonymous with "non-computable". The two meanings of "undecidable" do not coincide, and don't even apply to the same kind of things. In particular, "computable" doesn't apply to single sentences at all.)
But what about unprovable sentences that are not undecidable? Can they also be added as axioms?
Such sentences are necessarily disprovable (because if they were not, they would be undecidable). So if we try to add them as axioms, we get an inconsistent theory.
Now, do phrases like "In my opinion [unprovable sentence here] is true" make sense?
Yes, if they are taken to be about truth in the "intended model" of whichever language your theory is phrased in, such as the actual integers. Most of us feel intuitively that the integers exist in some Platonic way, independently of any formal systems for reasoning about them, and that all sentences in the language of arithmetic have a definite (though not necessarily known) truth value when applied to the actual integers.
Axioms
Originally, "axioms" meant "self-evident truths", or at least what seemed self-evident. But the more important question is what axioms are used for. From the beginning, logic in some form has been an essential part of reasoning, and we reason about things all the time. Then whenever we want to convey our reasoning to other people, and want them to accept our viewpoint, we need to present a valid argument. What is a valid argument? It is a series of inferences or deductions, each one following from the previous ones logically. But no argument can get started without first making at least one assumption. If the other person accepts all our assumptions as well as all the deductive steps we take, they would also have to accept the conclusion of our argument.
The goal in convincing another person, therefore, is to make as few and as weak assumptions as we can in our argument. Sometimes, there are assumptions that the vast majority of people accept, in which case we could call them self-evident truths. This is a common source of the assumptions that we freely use in everyday argumentation.
But sometimes we do not want to bother ourselves with small details of the real world that might have to be catered to in self-evident truths. In this case, we often come up with idealized forms, that we still call axioms. The axioms that Euclid came up with to describe geometrical objects are of this kind. Although there are no physical manifestations of ideal lines (with zero thickness) and ideal points (with zero diameter) and ideal circles, what we can derive about ideal lines and points and circles is so general that the slight deviation of physical entities from these ideal objects does not affect the vast utility of the mathematical theorems about the ideal geometrical world. All we need to remember is that the ideal world is merely an approximation of the real world, and we can effectively apply the theorems (such as Pythagoras' theorem) to reality by taking those deviations into account.
Approximation is but the first step in from the real to the abstract. The objective of abstraction is to attempt to isolate the crucial structures from the not so important details or specific data. The next step is to consider worlds governed by different axioms (assumptions). A well-known historical example was the exploration of geometrical worlds that satisfy Euclid's axioms minus the parallel postulate. In that case, there are ways to interpret some of these worlds in a Euclidean world. But the point of laying down a collection of axioms describing a world is so that we can thereafter argue completely based on deductive reasoning without any appeal whatsoever to the intuitive nature of the axioms (whether or not they are)! This is the modern meaning of "axiom" in mathematics, more or less.
Algebraic structures
Now about your questions regarding the axiomatization of groups, first let us see what it really is about. The axioms for a group govern a single group, a world in which there is only one binary operation and it satisfies those properties as specified by the axioms. Inside any such world, you cannot tell whether it is finite or not, for example, not to say know the size of the world. Outside the world, however, you can, as long as your outside world is strong enough to talk about worlds that satisfy the group axioms!
In logic, a world satisfying a collection of axioms is called a model. Commonly, we work in a higher world that satisfies the axioms of ZFC, in which we can talk about sets of axioms and sets that are models for some set of axioms. That is precisely why we are able to prove Lagrange's theorem; we cannot express it as a statement over the language of group theory, as you noted, which intuitively you can understand to mean that if you are 'inside' any group you do not even have the ability to express the theorem, not to say prove it.
There are indeed many more theorems you can prove in ZFC about groups than you can simply by deduction using only the group axioms. This is a very common phenomenon in mathematics, where you need to work from 'outside' all algebraic structures of a certain family (such as fields) to do anything much. The reason is that the axioms governing the algebraic structure hold for every member of the family that satisfy those axioms, so naturally you are very limited in what you can prove using them.
Best Answer
Axioms are the formalizations of notions and ideas into mathematics. They don't come from nowhere, they come from taking a concrete object, in a certain context and trying to make it abstract.
You start by working with a concrete object. After some investigation, you distill the axioms from that concrete objects, often you make mistakes and you need to add, or you can remove, some of the axioms.
For example, you work with the real numbers. But then you realize that most of what you need is in fact just a field which satisfies certain properties. These become axioms, for example, real-closed fields.
Or another example, you work with "collections" and you name them sets, at first they are just sets of numbers or sets of functions, but then you realize that you can talk about sets of these sets and sets of sets of these sets, and so on. You work and work with them, and you get some general idea about what properties these "sets" should have. For example, if you have a set, then collection of all its subsets should also be a set. So you write this formally and you have the power set axiom. And slowly you shape axioms for set theory.