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.
Caveat: in order to give you an overview, I've been vague/sloppy in several places.
Well the basic link to representation theory is that modular forms (and automorphic forms) can be viewed as functions in representation spaces of reductive groups. What I mean is the following: take for example a modular form, i.e. a function $f$ on the upper-half plane satisfying certain conditions. Since the upper-half plane is a quotient of $G=\mathrm{GL}(2,\mathbf{R})$, you can pull $f$ back to a function on $G$ (technically you massage it a bit, but this is the main idea) which will be invariant under a discrete subgroup $\Gamma$. Functions that look like this are called automorphic forms on $G$. The space all automorphic forms on $G$ is a representation of $G$ (via the right regular represenation, i.e. $(gf)(x)=f(xg)$). Basically, any irreducible subrepresentation of the space of automorphic forms is what is called an automorphic representation of $G$. So, modular forms can be viewed as certain vectors in certain (generally infinite-dimensional) representations of $G$. In this context, one can define the Hecke algebra of $G$ as the complex-valued $C^\infty$ functions on $G$ with compact support viewed as a ring under convolution. This is a substitute for the group ring that occurs in the representation theory of finite groups, i.e. the (possibly infinite-dimensional) group representations of $G$ should correspond to the (possibly infinite-dimensional) algebra representations of its Hecke algebra. This type of stuff is the basic connection of modular forms to representation theory and it goes back at least to Gelfand–Graev–Piatestkii-Shapiro's Representation theory and automorphic functions. You can replace $G$ with a general reductive group.
To get to more advanced stuff, you need to start viewing modular forms not just as functions on $\mathrm{GL}(2,\mathbf{R})$ but rather on $\mathrm{GL}(2,\mathbf{A})$, where $\mathbf{A}$ are the adeles of $\mathbf{Q}$. This is a "restricted direct product" of $\mathrm{GL}(2,\mathbf{R})$ and $\mathrm{GL}(2,\mathbf{Q}_p)$ for all primes $p$. Again you can define a Hecke algebra. It will break up into a "restricted tensor product" of the local Hecke algebras as $H=\otimes_v^\prime H_v$ where $v$ runs over all primes $p$ and $\infty$ ($\infty$ is the infinite prime and corresponds to $\mathbf{R}$). For a prime $p$, $H_p$ is the space of locally constant compact support complex-valued functions on the double-coset space $K\backslash\mathrm{GL}(2,\mathbf{Q}_p)/K$ where $K$ is the maximal compact subgroup $\mathrm{GL}(2,\mathbf{Z}_p)$. If you take something like the characteristic function of the double coset $KA_pK$ where $A_p$ is the matrix with $p$ and $1$ down the diagonal, and look at how to acts on a modular form you'll see that this is the Hecke operator $T_p$.
Then there's the connection with number theory. This is mostly encompassed under the phrase "Langlands program" and is a significantly more complicated beast than the above stuff. At least part of this started with Langlands classification of the admissible representation of real reductive groups. He noticed that he could phrase the parametrization of the admissible representations say of $\mathrm{GL}(n,\mathbf{R})$ in a way that made sense for $\mathrm{GL}(n,\mathbf{Q}_p)$. This sets up a (conjectural, though known now for $\mathrm{GL}(n)$) correspondence between admissible representations of $\mathrm{GL}(n,\mathbf{Q}_p)$ and certain $n$-dimensional representations of a group that's related to the absolute Galois group of $\mathbf{Q}_p$ (the Weil–Deligne group). This is called the Local Langlands Correspondence. The Global Langlands Correspondence is that a similar kind of relation holds between automorphic representations of $\mathrm{GL}(n,\mathbf{A})$ and $n$-dimensional representations of some group related to Galois group (the conjectural Langlands group). These correspondences should be nice in that things that happen on one side should correspond to things happening on the other. This fits into another part of the Langlands program which is the functoriality conjectures (really the correspondences are special cases). Basically, if you have two reductive groups $G$ and $H$ and a certain type of map from one to the other, then you should be able to transfer automorphic representations from one to the other. From this view point, the algebraic geometry side of the picture enters simply as the source for proving instances of the Langlands conjectures. Pretty much the only way to take an automorphic representation and prove that it has an associated Galois representation is to construct a geometric object whose cohomology has both an action of the Hecke algebra and the Galois group and decompose it into pieces and pick out the one you want.
As for suggestions on what to read, I found Gelbart's book Automorphic forms on adele groups pretty readable. This will get you through some of what I've written in the first two paragraphs for the group $\mathrm{GL}(2)$. The most comprehensive reference is the Corvallis proceedings available freely at ams.org. To get into the Langlands program there's the book an introduction to the Langlands program (google books) you could look at. It's really a vast subject and I didn't learn from any one or few sources. But hopefully what I've written has helped you out a bit. I think I need to go to bed now. G'night.
Best Answer
The lattices corresponding to elliptic curves are not the same as the lattices appearing in Lie theory. In the theory of modular forms, it is the weight $k$ (lying in the lattice $\mathbb Z$) that is a manifestation of Lie theory. (The lattice $\mathbb Z$ is the weight lattice of $SO(2)$.)
Added: Davidac897 asks "Why are you so sure"? Of course, there could be a connection that I am missing; I am just speaking from my own experience with these objects. But in the passage from classical modular forms (say as described in Serre's Course in arithmetic) to the representation-theoretic point of view, a lattice $\Lambda$ in $\mathbb C$ becomes a lattice $\mathbb \Lambda$ in $\mathbb R^2$, which becomes a point of $\mathrm{GL}_2(\mathbb Z)\backslash \mathrm{GL}_2(\mathbb R)$. Thus the lattices corresponding to elliptic curves become points of a quotient $\Gamma\backslash G$ for some real Lie group $G$ and some cofinite volume discrete subgroup $\Gamma$. For more general real Lie groups, one can think of such quotients as being a kind of generalized "moduli of lattices of some rank with some structure". There may be points corresponding to lattices with special symmetry (e.g. the square or triangular lattice in the elliptic curve case), and, yes, you are correct that root systems are lattices with special symmetry.
But this is not the usual way that root systems intervene in the theory of automorphic forms, which is what my answer above was intended to point out.