Generating functions are a useful mathematical tool which help analyze sequences $\{a_{n}\}_{n\in\omega}$ by taking the expression $a_{0}+a_{1}x+a_{2}x^{2}+\dots$ and doing things like multiplying by $x$, taking advantage of recurrences $a_{i}$ may satisfy, and so on.
My first question is this: what formally justifies this (if I may) "infinitary algebra" where we treat expressions like $1+x+x^{2}+\dots$ as we would finite expressions in the sense that we can add, subtract, multiply, and divide? Obviously in many cases we can write something of the form $\cfrac{1}{1-x}=1+x+x^{2}+\dots$, but what allows us to do algebra with this expression when we view $x$ as a "dummy variable" for the sake of some combinatoric/algebraic/other problem, while performing algebra in other contexts is prohibited? For example, letting $x=-1$ we have the infamous series $\cfrac{1}{1-(-1)}=\cfrac{1}{2}=1-1+1-1+\dots$. Elementary calculus says this is nonsense, as $\lim\limits_{n\rightarrow\infty}\sum\limits_{k=0}^{n}(-1)^{k}$ does not converge. Why should we then believe that $\cfrac{1}{1-x}$ is a good way of representing $1+x+x^{2}+\dots$ in the context of, say, some combinatorics problem we've decided to use generating functions to solve?
My second question is related to my above comment: "Elementary calculus says this is nonsense." But perhaps there is some advantage, then, to defining $1+x+x^{2}+\dots$ as $\cfrac{1}{1-x}$ instead of some limit. Indeed, this seems to be far more general as we can not only dispose of the limit (meaning we can analyze this expression in more general contexts like an arbitrary field), but even in contexts where the limit is already defined, like in $\mathbb{R}$, we can drastically expand the domain of this function (before, it was $(-1,1)$, and now it's $\mathbb{R}\setminus\{1\}$).
I would also like to point out that while most of the time the variable in a generating function serves as a dummy variable not meant to stand in for anything, there are times where it helps to actually give this a value.
In short: How do we formalize the manipulation of (infinite) generating functions in any context, but especially in an algebraic one? Is there an advantage to using the (often finite) representations of a generating function to define infinite sums?
Any reference on generating functions (particularly those where $\{a_{n}\}$ satisfies a linear recurrence) and their relationships with analysis or number theory is also welcome. This seems like a very interesting area and I would like to read more about it.