Question 1. When did the modern definition of a polynomial (as a sequence of coefficients, with multiplication defined by $\left(ab\right)_n = \sum\limits_{k=0}^n a_k b_{n-k}$) emerge?
Let me clarify: People have been working with some notion of polynomials for centuries, but mostly without a rigorous concept of what a polynomial is. From what I understand, "equations" rather than polynomials were the focus, and demarcating the precise extent of what an "equation" is was secondary to actually solving equations of given, fairly specific types. When the need for a rigorous definition became clear, it seems that the go-to solution was to define a polynomial as a polynomial function (e.g., §1 of Roland Weitzenböck, Invariantentheorie, 1923). This worked well for polynomials over $\mathbb R$ or $\mathbb C$, but not so much for polynomials over finite fields (which may be why E. H. Moore had to define $\mathbb{F}_{p^n}$ as a quotient ring of a polynomial ring over $\mathbb Z$ rather than of a polynomial ring over $\mathbb{F}_p$ in his 1893 talk).
What I call the "modern definition" of a polynomial is one of the following two definitions:
either as an infinite sequence $\left(a_0, a_1, a_2, \ldots\right)$ of coefficients (with all but finitely many $n$ satisfying $a_n = 0$), with addition defined entrywise and multiplication defined by the $\left(ab\right)_n = \sum\limits_{k=0}^n a_k b_{n-k}$ formula (and the indeterminate defined as the sequence $\left(0,1,0,0,0,\ldots\right)$ rather than being some mythic "symbol" or "indeterminate value" or "variable"),
or as the monoid ring of the additive monoid $\left\{0,1,2,\ldots\right\}$.
The second of these two definitions appears in §15 of B. L. van der Waerden, Moderne Algebra I, 2te Auflage 1937. (I can't check the 1st edition, as I don't have it.) This gives a lower bound on the age of the modern definition. As for an upper bound, I am perplexed. Nicolas Bourbaki's Elements of the History of Mathematics grazes the question on its page 82:
In all this research, the fields that arise consist of "concrete" elements in the sense of classical mathematics - (complex) numbers or functions of complex variables. 32 But already Kronecker, in 1882, takes full account of the fact (obscurely sensed by GaUBB and Galois) that "indeterminates" only play the role in his theory of the basis elements of an algebra, and not that of variables in the sense of Analysis ([186 a], v. II, pp. 339-340); and, in 1887, he develops this idea, linked to a vast programme that aims at nothing less than recasting all mathematics by rejecting all that cannot be reduced to algebraic operations on the integers. It is on this occasion that, taking up an idea of Cauchy ([56 a], (1), v. X, pp. 312 and 351) who had defined the field $\mathbb C$ of complex numbers as the field of remainders $\mathbb R\left[X\right]/\left(X^2+1\right)$, Kronecker shows how the theory of algebraic numbers is completely independent of the "fundamental theorem of algebra" and even of the theory of real numbers, all fields of algebraic numbers (of finite degree) being isomorphic to a field of remainders $\mathbb Q\left(X\right)/\left(f\right)$ ($f$ an irreducible polynomial over $\mathbb Q$)([186 a], v. 1111, pp. 211-240).
Following the references here, I see some signs of the modern definition slowly dawning in the 19th Century: [Note: "Polynomials" are often called "forms" before ca. the 1930s, particularly when they are homogeneous.]
I don't know what exactly Cauchy had done; the [56 a] reference goes to his Oeuvres, which I don't have.
The first Kronecker reference goes to his Grundzüge einer arithmetischen theorie der algebraischen Grössen (1882), where he talks of "veränderliche oder unbestimmte Grössen" but never constructs any.
J. Molk, Sur une notion qui comprend celle de la divisibilité et sur la théorie générale de l'élimination, Acta Math. 6 (1885), pp. 1--165, spends its section 2 (and maybe the whole Chapitre I?) arguing about the difference between variables and indeterminates. Alas, the argument appears to veer into philosophical terrain rather quickly, which puts it beyond my French comprehension skills; thus I have no idea what conclusion he actually comes to.
H. Weber, Die allgemeinen Grundlagen der Galois'schen Gleichungstheorie, Mathematische Annalen 43 (1893), pp. 521--549 tries to define polynomials in its §3 (after defining the notions of groups and fields in ways that are equivalent to their modern definitions). The definition pays obvious tribute to the idea that polynomials should not be functions; in particular it explicitly says that two polynomials are understood to be equal if and only if their respective coefficients are equal (rather than their values being equal). However, a vague notion of "expressions of the form $\Phi\left(x,y,z,\ldots\right) = \sum a x^r y^s z^t \cdots$" underlies this definition, and (e.g.) the multiplication is never explicitly defined (nor its associativity proven). (To this day, some algebra textbooks geared towards non-math majors work on this level.)
So my question is: Who completed this definition? Weber (later)? Kronecker? Dedekind? Noether? van der Waerden? And while at that:
Question 2. Who defined formal power series?
Question 3. Who defined polynomials in noncommutative indeterminates?
These are litmus tests for Question 1; a rigorous definition of polynomials needs only a few trivial modifications to define formal power series, and a definition of noncommutative polynomials is not far away either.
The concept of a polynomial is, in a way, similar to the number $0$: It's an idea that, on its own, appears pedantic and meatless. But once it is established, it becomes a building block of a whole discipline that no one would want to be missing. (19th Century algebraists like Sylvester tend to use the unique factorization property of a polynomial ring without ever explaining what it is that they are factoring; from a modern perspective, this is a gaping hole in their proofs.) My hope is that, as this specific matter is much younger than the number $0$, we may have better clues to its authors.
