Are there limitations to notation? I know that there are certain rules that allow to introduce new symbols to a language, which can be found here: https://en.wikipedia.org/wiki/Extension_by_definitions or How could we formalize the introduction of new notation?. Now, reading this gives the impression that there are strict rules for introducing new notation. However, in books and papers that are written in other areas than logic, I think that notation is rather introduced "arbitrarily" rather than following these rules. I wonder, why this is "ok" to do, despite violating these "strict" rules.
Before encountering these articles, I thought that one is free in introducing new notation in the sense that one can pick any symbol to denote any object without respecting any rules whatsoever. Hence, given two sets $X,Y$ and a function $f:X \to Y$, as well as $x \in X$, I could, if I want, denote the object $f(x)$ as $a$, $\phi, \lambda^42131290$ or whatever, really. Another example could be: Suppose the natural number setting, then the peano axioms ensure that there exists an element, which will be denoted $0$. Also for all $n \in \mathbf{N}$ there exists an object $s(n) \in \mathbf{N}$. We denote $s(0)$ as $1$, but could technically also call it $2$, $a$, $+$ or whatever, really. Also $s(n)$ could just be called $k$ by suppressing the dependency on $n$, I suppose. Why are those not allowed?
In set theory, one also chooses notation in brackets $\{...\}$. Technically speaking, one should be able to choose anything, really, shouldn't one? Again, why is this restriced? I am aware that this notation is well established, but it is technically not necessary to choose exactly this notation, is it?
My questions can be summarized as: $(1)$ Can I call objects whatever I like?
$(2)$ Why are there restrictions to notation in formal logic and why can they be violated in (other areas) of maths?
$(3)$ Why are the rules chosen the way they are? I suppose they are this way to prevent mistakes. What can "go wrong" if one violates these?
$(4)$ In some stackexchange question that I can't find anymore, I have read that $:=$ is not a symbol that is allowed in logic. Why is that? One only renames objects, what can possibly go wrong?
Note: I already tried to touch on those points on another post I made, however the answers did not really mention answers to those questions, which is why I made this a question on its own.
 A: Generally: in mathematics, "logic" can mean "meta-logic that we use to derive results" (e.g. the logic in $x=7\implies x(x+1)=7\times 8=56$), or it can mean "the branch of mathematics that studies how rigid, formal rules can be used to make deductions in a systematic way". You link to a Wikipedia article on "Extension by definitions", a very specific topic within a particular branch of logic. It is not more widely applicable to notation in mathematics.
$(1)$ Can I call objects whatever I like?
Yes. Notation is language, and just as you can use whatever words you want, you can also use whatever notation you want - but in both cases, some explanation may be necessary, and some choices may be better than others. You will not be met with any more patience if you say "we introduce an arbitrary $\mathbb{R}\to\mathbb{R}$ function, $\lambda^{42131290}$" than if you tell someone "henceforth, I will refer to dogs as 'domesticated wolf descendants of the first kind'".
$(2)$ Why are there restrictions to notation in formal logic and why can they be violated in (other areas) of maths?
The point of formal logic is to assume as few things as possible and deduce as many things as possible. In any good article on logic, you will see as much informal notation as in any other area of mathematics - though the article will be mostly sentences and rarely contain extremely dense derivations laid out only with formal logic.
As such, Peano axioms do not need a $1$, so they do not have a $1$. They need $0$ and $S$, from which we can get a $1$: $S(0)$. If you are reading something about Peano arithmetic that mentions $1$, this is a meta-logical statement, not one within the system. If you define $k$ to refer to $s(n)$, this is not a statement in the language of Peano arithmetic, but a meta-logical statement used to convey to the reader something about the language of Peano arithmetic.
So, the premise of your question is flawed. It is not that notation is violated in "other areas" of maths. It is that strict notation is only required when speaking in a formal logical language.
(3) Why are the rules chosen the way they are? I suppose they are this way to prevent mistakes. What can "go wrong" if one violates these?
There is no answer to this without being more specific. You give the example of set notation $\{\cdots\}$. In this case, when you write $\{1,2,3,4\}$, it is immediately obvious to another mathematician what you mean. $\{\cdots\}$ is like a word in the English language: you don't have to define it, because the people you are speaking to know it already.
If there are reasons not to use it, then don't. What could "go wrong" is that your statement may become ambiguous or harder for another person to understand. When I see $1,2,3,4$, I think it is different to $2,1,3,4$. When I see $\{1,2,3,4\}$, I know that it is the same as $\{2,1,3,4\}$.
$(4)$ In some stackexchange question that I can't find anymore, I have read that $:=$ is not a symbol that is allowed in logic. Why is that? One only renames objects, what can possibly go wrong?
In some languages of formal logic, $:=$ is not allowed, just as we have seen that $1$ may not be allowed. The reason would be that the same deductions can be made from the logic without the symbol as with it, so it is omitted for simplicity and for neatness of the theory. In no branch of logic would you not be permitted to use the notation $:=$ to describe a fact about that logical language to another person.
