The first question is answered (modulo some details) by Forman in

R Forman, *Morse theory for cell complexes*. Advances in Mathematics, 134 pp 90 - 145, (1998).

Theorem 12.1 shows that a discrete Morse function $f$ on a polyhedron $X$ induces a discrete Morse function $\hat{f}$ on the subdivision $\hat{X}$ produced by bisecting a single $d$-cell $\sigma$ of dimension $d$ into two $d$-cells $\sigma_1$ and $\sigma_2$ which share a $d-1$-face $\tau$, and that (among other things) this $\hat{f}$ has precisely the same number of critical cells as $f$.

There are three things to check:

$X$ is collapsible if and only if it admits a discrete Morse function with precisely one critical cell (this is true).

Barycentric subdivision of $X$ may be achieved via a sequence of bisections so that each intermediate step produces a regular CW complex (I think this is true), and

The proof of this theorem does not require anything more from our polyhedron $X$ than the fact that it is a finite regular CW complex (this is true).