I don't think it is fruitful to think of the constants as representing "rules" of the universe. Rather they are parameters in the formulae that represent the rules. The only serious proposals that I am aware of for other universes existing is the "multiverse" ontology for quantum mechanics. Essentialy a particle is represented by a wave function psi(x), with x representing a location in space. So psi(x) provides a number, which is interpreted conventionaly as the probability of the particle being measured at location x. Integrating psi(x) over spacetime yields 1 because integrating a function is just summing it it's values at all points and the particle is someplace in the universe so the probabilities add to 1. In the multiverse ontology, we instead interpret psi(x) as the percentage of universes in which the particle is at location x. The evolution of psi(x) through time is governed by the schroedinger equation: ih(d/dt)psi(x) = Hpsi(x) (to my chagrin, i did have to look that up...). You'll notice the plank constant in there on the left side (the capital H is the hamiltonian, lets not get into that). So I would think that the planck constant, at least, would be constant across all universes (assuming these other universes exist, i'm somewhat dubious).
As far as your other questions go, they are fairly deep and I can only postulate.
1) A change in the parameters of the laws of physics would change the laws of physics: they don't change. yeah yeah, that only begs the question.
2) Within our current models, they are taken as givens (that is measured and then assumed in the models) or derived from other constants. We really don't know much about them other then that inertia has nothing to do with it.
3) When we figure out how to derive one from another or from formulae, one could say that we "understand" it.
Ultimately, these are difficult. Physics takes numbers representing measurements, runs them through formulae and returns numbers representing predictions. It does not say, how or why, only what, where and when.
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