Does the Higgs field really explain mass or just reformulate it? What about charge? 
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*The mass of a particle used to be considered a fundamental and intrinsic property of the particle; on the same level as other properties such as charge, spin, chirality/helicity. Due to the Higgs mechanism this has changed. The mass is now considered a property that is acquired as the result of interaction with the Higgs field. But why is this so relevant? Instead of an intrinsic mass, now each particle type has an intrinsic coupling constant which describes the strength of its interaction with the Higgs field. It seems to me that, as long as we have zero knowledge about the nature of this interaction and hence about its strength, the problem of mass has merely been reformulated into something equally mysterious.

*I suppose it may be of interest to propose a coupling of elementary particles to some background field that permeates the universe. One key question is then how this field strength can remain perfectly homogeneous in space and times. If no ripples are allowed, then there are no sources and sinks for the Higgs field. But how can it interact with matter, without being influenced by the very same interaction? This seems paradoxical to me. 

*Is there a particular reason why one can't propose a "Charge field", which is a quantum field that interacts with certain particles and thus gives them their Charge? 
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*An obvious difference between the two ways of thinking about it you mention is that in the case of the Higgs mechanism, there is an observable particle excitation of the field associated with it, which was found recently. Furthermore it should be noted that the Higgs mechanism only concerns the mass generation of some elementary particles. The mass of composite particles like hadrons is largely due to strong interactions, which are not related to the Higgs.   

*The idea of fields spanning the universe is not unique to the Higgs field. It is the principle that underlies quantum field theory which describes all particles of the standard model. I suggest that you acquire some profound knowledge on the subject in books by Zee or Srednicki. 

*The reason that mass is generated by the Higgs field can be traced back to the spontaneous breaking of a symmetry. There is no such symmetry that might explain the generation of charge in a similar way. 

A: I'll only address your first question here.
To start off with a sidenote, I think the idea that mass is a fundamental property of a particle has been on shaky ground ever since Einstein showed the equivalence of mass and energy. I can hardly imagine it took very long for people to come to the conclusion that mass cannot be a fundamental property of particles. Today we know how this mass energy comes about: basically in a similar way to how an electric charge gains energy in an electric field. One difference being that we have no control over the BEH field, while we do have some over electromagnetic fields.
So the change that has come with the discovery of the BEH boson is that we've climbed down a branch in the Fundamentality Tree$^1$. It's like living in a universe where a fairly strong, constant electric field is ever-present and you know from experience that it is harder to move some particles in one direction than it is in another.$^2$ So you ascribe a fundamental property to all particles that might be something like a directional mass. However you have no idea why this property exists. And then you discover the ever-present electric field: you find out that your directional mass is simply the consequence of particles coupling to that field with some strength.
In this hypothetical universe, the BEH field might not be ever-present with a non-zero vacuum expectation value. Say it could, however, be straightforwardly induced in some way. The hypothetical observers in this hypothetical universe would then straightforwardly discover the BEH field coupling and the energy gain of particles associated with moving through it. They could immediately associate coupling strengths with particles.
In our universe, coupling to the electromagnetic field is much clearer and we can directly observe the coupling strengths of different particles. Contrary to the hypothetical universe from before, coupling to the BEH field is the less obvious one in ours. So now we are at the same height in the Fundamentality Tree for mass energy as we are for e.g. electromagnetic energy: the level of coupling constants. These are equally mysterious for the BEH field as they are for the EM field, in the sense that we don't know why they have the values that they do.

$^1$ I imagine the Fundamentality Tree looks a bit like this.
$^2$ This is a completely alien universe to us of course, as charged particles would be accelerated constantly but that is besides the point. Let's also leave out of consideration whether or not life could exist in this universe... And whether the universe itself could actually exist.
A: This isn't a complete answer to your question, just two comments on two specific points that came up that were too long for a comment:


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*WRT the Higgs having a spatially varying background because of inhomogeneities in the universe: It's important to keep in mind that the Higgs field is VERY MASSIVE when it has a nonzero VEV (as it does in the universe today). So even though, for instance, a planet could in principle create a spatially varying higgs field background [which you can compute classically to a good approximation], the range of the "Higgs force" is something like 10^{-17}m, so the background $H(r) \sim e^{-mr}/r$ is cutoff over a very short distance indeed leaving the Higgs field essentially constant.

*WRT the vacuum energy of the Higgs: Indeed! The Higgs could in principle contribute to the vacuum energy and change the cosmological constant, this is one aspect of the cosmological constant problem! Indeed one of the original problems people like Anderson had with the Higgs mechanism was that it would give too large a classical contribution to the cosmological constant to be consistent with observations. Weinberg's perspective (in his classic paper on the cosmological constant problem) is that today we can think of the vacuum energy of the higgs field as being 0 (i.e.: $V(H)=0$ when $H=\langle H\rangle$, where $\langle H\rangle $ is the higgs vet today). This implies there was a larger cosmological constant in the past, when $H=0$! However, in the past there was also a higher energy density in matter and in radiation, and the contribution from these two pieces swamps the extra classical contribution to the cosmological constant from the Higgs field.
