For commutative rings $R \subseteq S$, recall that $S$ is separable over $R$, if $S$ is a projective $S \otimes_R S$-module, via $f: S \otimes_R S \to S$ given by: $f(s_1 \otimes_R s_2)=s_1s_2$.

Question 1: Is $\mathbb{C}[x]$ separable over $\mathbb{C}[x^2,x^3]$?

More generally,

Question 2: Is it possible to characterize all $\mathbb{C}$-subalgebras $\mathbb{C} \subset R \subset \mathbb{C}[x]$ such that $\mathbb{C}[x]$ is separable over $R$?

According to wikipedia: "Moreover, an algebra $S$ is separable if and only if it is flat when considered as a right module of $S \otimes_R S$ in the usual way". Here $\mathbb{C}[x^2,x^3] \subset \mathbb{C}[x]$ is not flat; I am not sure if there is a connection between flatness or non-flatness of $\mathbb{C}[x^2,x^3] \subset \mathbb{C}[x]$ and $\mathbb{C}[x] \otimes_{\mathbb{C}[x^2,x^3]} \mathbb{C}[x] \subset \mathbb{C}[x]$.

If, for example, $\mathbb{C}[x^2,x^3] \subset \mathbb{C}[x] \otimes_{\mathbb{C}[x^2,x^3]} \mathbb{C}[x]$ is flat (I do not know if this is true or false), then flatness of $\mathbb{C}[x] \otimes_{\mathbb{C}[x^2,x^3]} \mathbb{C}[x] \subset \mathbb{C}[x]$ would imply flatness of $\mathbb{C}[x^2,x^3] \subset \mathbb{C}[x]$, which is false.

(This question may be relevant. Also asked in MSE).

Edit: After receiving a comment that "it's unlikely that you can characterise all $R$ for which $R \subseteq \mathbb{C}[x]$ is separable, I would like to change Question 2 to the following question:

Question 3: Is it possible to characterize all $h \in \mathbb{C}[x]$, such that $\mathbb{C}[x]$ is separable over:

(i) $A=\mathbb{C}[h]$.

(ii) $B=\mathbb{C}+(h)$, where $(h)$ denotes the ideal of $\mathbb{C}[x]$ generated by $h$.

Example: If $h=x^2$, then $B=\mathbb{C}+(x^2)=\mathbb{C}[x^2,x^3] \subseteq \mathbb{C}[x]$ is not separable (first comment below).

Remark: Denote $h(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0$.

According to Corollary 8 with $h(Z)=c_nZ^n+c_{n-1}Z^{n-1}+\cdots+c_1Z+c_0-h$, we obtain an answer to Question 3(i): $\mathbb{C}[x]$ is separable over $\mathbb{C}[h]$ iff $\deg(h)=1$ (namely, $A=\mathbb{C}[x]$).

Still, I am not sure what can be said about Question 3(ii); any help is welcome!

Thank you very much!