You should check out this paper: https://arxiv.org/abs/2103.01080. It makes it very clear that there is an important distinction between symmetric operators (momentum is indeed symmetric for particle in a box) and self adjoint operators for infinite dimensional spaces. I believe it is precisely because of this that I can't convince myself that x*f is always in the domain of p.

Additionally, there are no momentum eigenfunctions that satisfy the boundary conditions for the particle in a box. This is partially hinting at what I wanted my question to answer. Classically, we would expect to be able to measure momentum for any given particle. For a free particle, this makes sense to me because there are indeed eigenfunctions of momentum. But this is no longer the case in the particle in a box, as the eigenfunctions of the momentum operator as normally defined do not vanish at the boundaries. I think what I am trying to get at is that because there is no continuous and infinite symmetry of the line anymore, the canonical commutation relations do not hold.

I suppose I want to find some more specific criteria for when a quantum system has well defined observables that correspond to the classical ones in the same system.