Renormalization scheme independence of beta function

I have some questions about renormalization. To my understanding, in order to deal with infinities that appear in loop integrals, one introduces some kind of regulator (eg, high momentum cutoff, taking $d\to d+\epsilon$, etc.) so that we get a finite answer, which blows up as we remove the regulator. Then we renormalize various coefficients in the Lagrangian in a regulator-dependent way so that, in scattering amplitudes, a finite piece remains as we remove the cutoff. The regulator seems to always require introducing some arbitrary scale (eg, the momentum cutoff or $\mu^\epsilon$ for dim-reg), although I'm not sure why this must be the case.

Now, this finite piece is completely arbitrary, and depends on what scheme we want to use (eg, on-shell, minimal subtraction, etc). The beta function is then, roughly speaking, the rate of change of this finite piece under changes in the scale in the regulator. My question is, what exactly is the invariant information in the beta function? Under changing renormalization scheme, it obviously changes, but it seems that this roughly corresponds to a diffeomorphism on the space of couplings (is this always true? for example, in on-shell renormalization, if we set the mass to the physical mass and the coupling to the corresponding exact vertex function at zero momentum, these seem to be independent of scale - do the beta functions vanish here?). Also, it seems to depend on exactly what regulator we are using, and how the dimensionful scale enters into it. There are certain quantities, such as the anomalous dimension of a field at a fixed point, which must be independent of all these choices, but I don't understand why this is the case.

Finally, and this is more of a philosophical question, what exactly do the infinities in the loop integrals mean in the first place? Why is it that we can remove them by any of these various regulators and expect to get the same answer? I've read something about Haag's theorem, that interacting theories live in a different Hilbert space than the free-field Fock space, and that this is somehow related, but I'm not sure why. Thank you.