  Symmetric functions appear in many areas of mathematics and physics,
including enumerative combinatorics, the representation theory of symmetric
groups, statistical mechanics, and the quantum statistics of ideal gases. In
the commutative (or "even") case of these symmetric functions, Kostant and
Kumar introduced a nilHecke algebra that categorifies the quantum group
$U_q(\mathfrak{sl}_2)$. This categorification helps to better understand
Khovanov homology, which has important applications in studying knot
polynomials and gauge theory. Recently, Ellis and Khovanov initiated the
program of "oddification" as an effort to create a representation theoretic
understanding of a new "odd" Khovanov homology, which often yields more
powerful results than regular Khovanov homology. In this paper, we contribute
towards the project of oddification by studying the odd Dunkl operators of
Khongsap and Wang in the setting of the odd nilHecke algebra. Specifically, we
show that odd divided difference operators can be used to construct odd Dunkl
operators, which we use to give a representation of $\mathfrak{sl}_2$ on the
algebra of skew polynomials and evaluate the odd Dunkl Laplacian. We then
investigate $q$-analogs of divided difference operators to introduce new
algebras that are similar to the even and odd nilHecke algebras and act on
$q$-symmetric polynomials. We describe such algebras for all previously
unstudied values of $q$. We conclude by generalizing a diagrammatic method and
developing the novel method of insertion in order to study $q$-symmetric
polynomials from the perspective of bialgebras.
