To start with part 3: "local-global principles" of various kinds are one of the big themes in number theory, at least. Starting with, for example, a positive integer k being a square if and only if it is a square modulo all primes. That doesn't explicitly use local fields; but extensions to the idea, going under the general name of "Hasse principle", do use local fields in their formulation. A major effort in Diophantine equations, for the theory of existence of rational solutions, has been to understand when the Hasse principle holds; and when it doesn't to explain how to modify it.
To answer 1 with an example: class field theory is much easier in the local case, and gives a relatively slick theory. Again the approach is associated with the name of Hasse. When you move to "non-abelian class field theory", a.k.a. the Langlands philosophy, local fields are part of the basic formulation (adelic).
My attitude to 2 is that we don't really know the scope, and that is part of the jury being out on "number theory". There are "p-adic analogues" of many things. My advisor used to say that it was sheer prejudice and force of habit that the real numbers were the first local field taught. That was a joke, but there is a grain of truth in it. It is probably helpful to see the "empire building" of local fields as coming out of classical techniques that work for quadratic forms and cyclotomic fields (Kummer) and have contributed to many other areas by now, in different ways (e.g. non-archimedean Lie groups).