This question is put on hold although Deane Yang, who did this, has answers to issues like "Why do we teach calculus students the derivative as a limit?" - how's the latter research, actually?

Dear @Ricardo Andrade, thanks for your feedback. As you suggested, I added the references.

Yes, of course. For example, the number $S(n,k)$ from above is precisely the number $d_{S,f}(n,k)$ defined in Eger (2013), Restricted weighted integer compositions and extended binomial coefficients (for $S=\{1,2,3,\ldots,\}$ and $f(a)=a^p$). This paper says that $S(n,k)$ is an extended binomial coefficient, and gives various representations of the extended binomial coefficients. Other relevant literature would be Fahssi (2012), The polynomial triangles revisited, and Shapcott (2013), C-color compositions and palindromes. More relevant literature can be found in the references of these works.