Let $X$ be a path-connected CW-complex, with a CW-structure with one $0$-cell.
We define the cone on $X$ to be the space
$$CX:= (X \times I) /(X \times \{1\}).$$
If we equip the interval $I$ with the usual CW-structure with two $0$-cells and one $1$-cell, then the CW-structure which is induced on $CX$ has two $0$-cells. One corresponds to the unique $0$-cell in $X$, and one corresponds to the factored-out $X \times \{1\}$. $X$ embeds as a CW-complex into $CX$.
Is it possible to construct the cone on $X$ in such a way that it can be equipped with a CW-structure with only one $0$-cell, and such that $X \hookrightarrow CX$ is still an inclusion of $CW$-complexes? Or, is there an alternative way of embedding $X$ - as a CW-complex - into a contractible space $CX$ which has only one zero cell (this contractible space not having to be homeomorphic to the cone)? A sort of homotopy theoretic analogue to the above construction, but with this alternative CW-structure?
This is possible in the case of $X = S^1$, for example. We can equip $CX = D^2$ with a structure where we have "squashed together" the two $0$-cells which would arise after constructing the cone on $S^1$ in the above way.
Yes! This can be gratuitously called a model for the 0th Postnikov approximation of $X$.
Start with $X$ and inductively attach cells to kill all the homotopy groups, similarly as to how you would create an Eilenberg-Maclane space. After you have killed all the homotopy groups, we have a CW complex with trivial homotopy groups. By Whitehead, this is contractible. By construction, X is a subcomplex. Clearly we have not added any 0-cells.
An interesting question is whether you can make this functorial without changing the requirement on 0-cells.