I wasn't sure how to formulate the title.
Say I have a space $\mathbb R^3$ with the coordinates $(x_1,x_2,x_3)$. Does it mean ALL functions in this $\mathbb R^3$ must have the coordinates $(x_1,x_2,x_3)$?
I mean, say I have the functions $f,g:\mathbb R^3\rightarrow\mathbb R$, does it mean they have the coordinates $f(x_1,x_2,x_3)$ and $g(x_1,x_2,x_3)$?
Or could they have different coordinates in, say $f(x_1,x_2,x_3)$ and $g(x,y,z)$?
Do I have to be consistent with the notation of the coordinates?
Q2: Say I want to introduce the function $h:\mathbb R\rightarrow\mathbb R^3$, does i mean I have to use $x_1$ as the input $h(x_1)$?
You don't have to, but you should.
Like all language, mathematical notation is all about communication. The point of writing mathematics down is to move ideas from your head to someone elses head (or possibly back into your own head at a later time). To achieve this, you write down your ideas using a mutually understood language. Some of it will use English, or whatever other human language you and your reader prefer to use, and some of it will use mathematical notation.
In order to fulfill this purpose, it is desireable that whatever you write is as easy as possible to read. Part of that is to make sure that the reader doesn't have to keep track of too much at the same time, and doesn't get overloaded with new stuff to keep track of at every turn, as well as to play along with the reader's expectations.
If you have written about a function $f:\Bbb R^3\to \Bbb R$, that takes the three variables $x_1, x_2, x_3$ as input, and you then introduce another function $g:\Bbb R^3\to \Bbb R$, then if you describe its inputs as $x, y, z$, well, that's another set of symbols that your reader now has to remember and keep track of as they read. At the same time, they will wonder whether there was any reason to switch, and what will happen with $x_1, x_2, x_3$. All this goes on in the back of their minds, and distract them from being able to follow your actual argument. At least, it would do that to me.