There are no known applications and no imaginable applications of the Higgs boson which is not surprising given the fact that its lifetime is a zeptosecond.
The Higgs boson is a particular particle – state in the Hilbert space – in a correct quantum mechanical theory describing Nature. I mean the Standard Model or its extensions. Any discussion of the Higgs boson would be totally impossible without quantum mechanics. All properties of the Higgs boson crucially depend on principles and special effects of quantum mechanics.
The Higgs boson is a particle associated with the Higgs field. To see the emergence of particles from fields, one has to discuss physics at the level of quantum mechanics; see the previous point. However, even in classical physics, one may add the Higgs field to the general theory of relativity, much like the electromagnetic fields. The Higgs field is a source of gravity and other things. But it's just "another added player"; the main field in the general theory of relativity is the metric tensor, i.e. the spacetime geometry, not the Higgs field.
All realistic models of string theory that are or were trying to describe the Universe around us contain a Higgs field and a Higgs boson; string theory seems incompatible with all the alternatives. In this sense, the discovery of the Higgs is a victory for string theory and a confirmation of one of its predictions (a relatively uncontroversial prediction). Scalar fields – a broader family of fields similar to the Higgs field (fields without spin) – are also generic and important in string theory, see
I think that none of the points you asked answers the question in the title, "what's next". We don't know what will be the next discovery. Physics researcher is not Stalin's five-year plan. If we knew what the next breakthrough would be, we would already to it tonight. One may only discuss what ideas people are studying in their effort to make the big breakthrough. When it comes to physics that is as analogous to the Higgs boson as possible, supersymmetry would probably be the top pick.