First of all, Kevin is being quite modest in his comment above: his paper
Buzzard, Kevin. Integral models of certain Shimura curves. Duke Math. J. 87 (1997), no. 3, 591--612.
contains many basic results on integral models of Shimura curves over totally real fields, and is widely cited by workers in the field: 22 citations on MathSciNet. The most recent is a paper of mine:
Clark, Pete L. On the Hasse principle for Shimura curves. Israel J. Math. 171 (2009), 349--365.
http://alpha.math.uga.edu/~pete/plclarkarxiv7.pdf
Section 3 of this paper spends 2-3 pages summarizing results on the structure of the canonical integral model of a Shimura curve over $\mathbb{Q}$ (with applications to the existence of local points). From the introduction to this paper:
"This result [something about local points] follows readily enough from a description of their [certain Shimura curves over Q] integral canonical models. Unfortunately I know of no unique, complete reference for this material. I have myself written first (my 2003 Harvard thesis) and second (notes from a 2005 ISM course in Montreal) approximations of such a work, and in so doing I have come to respect the difficulty of this expository problem."
I wrote that about three years ago, and I still feel that way today. Here are the documents:
- http://alpha.math.uga.edu/~pete/thesis.pdf
is my thesis. "Chapter 0" is an exposition on Shimura curves: it is about 50 pages long.
- For my (incomplete) lecture notes from 2005, go to
http://alpha.math.uga.edu/~pete/expositions2012.html
and scroll down to "Shimura Curves". There are 12 files there, totalling 106 pages [perhaps I should also compile them into a single file]. On the other hand, the title of the course was Shimura Varieties, and although I don't so much as attempt to give the definition of a general Shimura variety, some of the discussion includes other PEL-type Shimura varieties like Hilbert and Siegel moduli space. These notes do not entirely supercede my thesis: each contains some material that the other omits.
When I applied for an NSF grant 3 years ago, I mentioned that if I got the grant, as part of my larger impact I would write a book on Shimura curves. Three years later I have written up some new material (as yet unreleased) but am wishing that I had not said that so directly: I would need at least a full semester off to make real progress (partly, of course, to better understand much of the material).
Let me explain the scope of the problem as follows: there does not even exist a single, reasonably comprehensive reference on the arithmetic geometry of the classical modular curves (i.e., $X_0(N)$ and such). This would-be bible of modular curves ought to contain most of the material from Shimura's book (260 pages) and the book of Katz and Mazur Arithmetic Moduli of Elliptic Curves (514 pages). These two books don't mess around and have little overlap, so you get a lower bound of, say, 700 pages that way.
Conversely, I claim that there is some reasonable topology on the arithmetic geometry of modular curves whose compactification is the theory of Shimura curves. The reason is that in many cases there are several ways to establish a result about modular curves, and "the right one" generalizes to Shimura curves with little trouble. (For example, to define the rational canonical model for classical modular curves, one could use the theory of Fourier expansions at the cusps -- which won't generalize -- or the theory of moduli spaces -- which generalizes immediately. Better yet is to use Shimura's theory of special points, which nowadays you need to know anyway to study Heegner point constructions.) Most of the remainder concerns quaternion arithmetic, which, while technical, is nowadays well understood and worked out.
Your questions would require an enormous amount of work to answer properly, so let me just suggest a few modest and very partial answers to your 1)2)3).
1) Modular forms are shiny: they satisfy or explain many beautiful and surprising numerical identities (about partitions and sums of square among others). This got them noticed in the first place.
2) Modular forms have Galois representations, and conversely Galois representations often come from modular forms. If you care at all about representations of the absolute Galois group of $\mathbb Q$, then you will first presumably be interested in class field theory, and develop the Kronecker-Weber theorem. But then you will get interested in representations of $G_{\mathbb Q}$ of rank 2. Modular forms provide many examples of such Galois representations, and conversely, only a handful of hypotheses are required for such a Galois representation to come from a modular form. This means concretely that one can identify many Galois representations simply by computing a few traces of Frobenius morphisms and then doing some computations in the complex upper half-plane.
3) If a rational elliptic curve has a non-vanishing $L$-function at 1, it has no non-torsion rational points. The main conjecture of Iwasawa (about class groups in the cyclotomic $\mathbb Z_{p}$-extension of $\mathbb Q$) is true. Fermat's last theorem is true. Here are three extremely famous conjectures solved by an ubiquitous appeal to modular forms. All these conjectures were well-known in the 60s but I don't think it is an exaggeration to say that almost no one then would have suspected that modular forms would come into play.
Best Answer
I don't know what you mean by Modular forms of moduli stack, I think maybe you mean modular forms on moduli stacks. Either way, you should probably have a look at the book by Katz--Mazur titled "Arithmetic Moduli of Elliptic Curves". It should explain how to think about modular curves in the correct setting you want. In order to understand the modular forms rephrased in this language, the reference to Katz's paper "p-adic properties..." is a really good one, but nowadays there are lots of notes online which explain the "Katz definition of modular forms". I think some really good notes on this are Frank Calegari's AWS notes which you can find here: http://swc.math.arizona.edu/aws/2013/ They should take you from the modular forms you know and love to Katz's version.
Also, if its your first time seeing stacks and are scared of such things (like I am) then you are probably fine just ignoring the word stack and just think about a scheme, since, in most cases you can just rig it so that the moduli problem is representable by some scheme (i.e. by making sure you have a nice enough level (a crucial term being sufficiently small))
Lastly, I don't know what you kids these days mean by main stream, but looking at books like Cornell--Silverman--Stevens on the proof of Fermat's Last Theorem would give you an idea of the popular tools are used in number theory and arithmetic geometry (although some might even say this is now old stuff and if you want to see the future then you need to look at things people like Scholze are doing..)