Nothing stops your function $f$, which is defined on all of $\mathbb{R}^d$, from having critical points in the interior of $X$. Morse theory is defined for manifolds with boundary (and more generally, for Whitney stratified spaces), but in all the cases that I know of there will generally be critical points in $X - \partial X$. Could you specify why you need $X$ at all, rather just defining a standard Morse function on $\partial X$?. In any event, see arxiv.org/abs/1207.3066

I'm confused; if $Y$ is smooth, shouldn't $\text{Sing}(Y)$ be empty, rather than equal to all of $Y$?

Thanks, @AndréHenriques! Serves me right for asking here before walking over to ask you or Chris :) In my case there is no unital structure to the two algebras, so I am saved from Eckmann Hilton through other means...

I will confess that I had to read Prop 1.6 about 5 times before I realized that the two sides of the equality were actually different! For others suffering similarly, note that $x_6$ and $x_7$ are swapped. (And yes, I was indeed aware of Eckmann-Hilton but none of my products are unital). I will wait a bit to see if anyone else has seen the structure and given it a catchy name, and will happily accept this answer otherwise :)

It is not true that $f^{-1}(0)$ is a smooth manifold unless $0$ is a regular value of $f$. Take any closed subset $C \subset \mathbb{R}^n$ and let $f:\mathbb{R}^n \to \mathbb{R}$ be the (smooth!) function sending $x$ to the squared Euclidean distance from $x$ to $C$. Then $C = f^{-1}(0)$ even though $C$ could be a horrible Cantor set.

@BenjaminSteinberg sorry, no. I managed to bypass the need for it after several failed attempts at googling