So I've just finished my first uni course on quantum physics, and one of the more beautiful moments for me was when I understood completely how the quantum state of a system is not defined in terms of its position wavefunction, or momentum wavefunction, or energy eigenvalues etc, but simply exists in an abstract vector space whose "components" (to speak loosely) are different depending on your choice of basis.

Nonetheless, some questions remain tied to the position basis. Specifically, why in position basis is the momentum operator negative i h-bar times del (or the partial derivative with respect to x, in one dimension). Why, both on a formally and intuitively, is this the case?

Additionally, why can we almost always slot the position and momentum operators into their respective positions in newtonian equations to get their equivalents?

Thank you, if you give an answer which is comprehensive but doesn't quite address what I'm attempting to express, I'll comment with questions and re-clarifications to help us get on the same page.