If a polynomial ring is a free module over some subring, is that subring itself a polynomial ring?

The third bullet point is automatic by Noether Normalization. What you ask is certainly true if you assume that $A=A_0\oplus A_1\oplus A_2\oplus\cdots$ as a graded ring with $A_0=k$ and $A_1$ generates it. This follows from the fact that your assumptions imply $A$ is a regular ring and then if $A$ is of the form I wrote, $\dim_k A_1=n$ and the rest will follow.

The regularity of $A$ follows from your hypothesis even without gradedness assumption. I do not offhand know an example where $A$ is not a polynomial ring.

The answer to my question is in the affirmative. I've now found two proofs. The result is apparently old and originally due to Macaulay.

One is on pp. 155 and 171 of Larry Smith's book Polynomial Invariants of Finite Groups. It uses some homological algebra to prove first that without the finiteness assumption, the polynomial ring is free over $A$ if and only if $A$ is a polynomial ring on a regular sequence of generators. More than this, finiteness of the polynomial ring over $A$ shows this sequence can have length no less than $n$, and regularity that it can have length no more than $n$. It is also shown that given a sequence $f_1,\ldots,f_n \in k[x_1,\ldots,x_n]$, it is regular if and only if the $f_i$ are algebraically independent and $k[x_1,\ldots,x_n]$ is a finite $k[f_1,\ldots,f_n]$-module.

The second is in Richard Kane's Reflection Groups and Invariant Theory. Starting on p. 195, he wants to show that if a polynomial ring $S(V)$ (symmetric algebra on a vector space $V$) is finite over $R = S(V)^G$ the invariants of some $G$-action by pseudo-reflections on $V$ and $S$ is a finite, free $R$-module, then $R$ is a polynomial algebra. He carries this through with a generators-and-relations level approach involving formal partial derivatives and Euler's theorem on homogeneous functions to leverage a hypothetical algebraic relation on the generators of $R$ into an $S$-linear and then an $R$-linear relation among these same generators and contradict minimality. But nothing in his approach uses anything other than that $S$ is a polynomial ring and there is some generator of $R$ of order relatively prime to the characteristic of $k$.