Differentiability on $\mathbb{R}^n$ under an equivalence relation.

It depends on the equivalence relation. Differentiability is defined in smooth manifolds, and any quotient of a smooth manifold may not be another manifold. From what you say it seems that you have studied manifolds inmersed in $\mathbb{R}^n$, but this seems like it could be better understood by studying manifolds in an abstract setting. A reference for that is $\textit{Introduction to Smooth Manifolds}$, by Lee.

The result you are interested in can be found in the 2$^{nd}$ edition of the book as theorem 21.13 (9.13 in the 1$^{st}$ edition). Your equivalence relation is given by a discrete group acting freely and properly in $\mathbb{S}^2$ (see example 21.14), so the quotient is a manifold (diffeomorphic to the projective plane, $\mathbb{P}\mathbb{R}^2$). This means that we have a smooth quotient map $\pi:\mathbb{S}^2\to\mathbb{S}/{\sim}$. In the case of discrete groups (like in your example), the map $\pi$ is a local diffeomorphism.

Now to answer the question, if a map $\mathbb{S}^2\to\mathbb{S}^2$ is smooth, then the map $\pi\circ f:\mathbb{S}^2\to\mathbb{S}/{\sim}$ is smooth (as it is a composition of smooth maps). Now as you say, the condition $f(x)=f(-x)\;\forall x$ means that from $\pi\circ f$ we can obtain a well defined function $g:\mathbb{S}/{\sim}\to\mathbb{S}/{\sim}$. Moreover smoothness can be checked locally, and locally the function $g$ is given by $f\circ\pi^{-1}$, which is differentiable because $\pi$ is a local diffeomorphism.

PS: $\textit{Introduction to Smooth Manifolds}$ is the main book I´ve used to study smooth manifolds but I don´t know if it´s the best reference for quotients of manifolds by group actions at this level, anyone with a better reference is welcome to edit the answer.