I haven't checked all the details but I think you can do this using the fact that $\binom{2n}{n}$ counts Dyck paths of length $2n$ in which up steps starting at height 0 are weighted by 2. This fact corresponds, via Flajolet's combinatorial approach to continued fractions, to the continued fraction $$ \frac{1}{\sqrt{1-4x}}= \cfrac{1}{1-\cfrac{2 x}{1-\cfrac{x}{1-\cfrac{x}{1-\cfrac{x}{1-\cfrac{x}{1-\dots}}}}}} $$

Some people use the word "combinatorial" for anything to do with (formal) power series. Other people call such things "analytic."

It might help if you defined Chebyshev transform.

There is a substantial literature on this problem. See, for example, djalil.chafai.net/blog/2012/05/03/….

For an explanation of the connection between $q$ as a variable and as a prime power in $q$-binomial coefficients, see D. E. Knuth's paper, Subspaces, subsets, and partitions, doi.org/10.1016/0097-3165(71)90022-7.

Dividing these numbers by $(2m)!$ gives the central factorial numbers of the second kind, oeis.org/A036969 and oeis.org/A008957.

$h_k[n\mathbf{z}]$ is the coefficient of $t^k$ in $(1+h_1t + h_2t^2+\cdots)^n$. This agrees with your examples for $k=n=4$ and $k=n=5$.