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2 Corrected typos

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 you will first presumably be interesting 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 representations 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 no one would have suspected that modular forms would come into play.

1 [made Community Wiki]

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 you will first presumably be interesting 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 representations 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. 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 no one would have suspected that modular forms would come into play.