As an answer to (2): I think the easiest to state application of modular forms to a domain not obviously connected to them is in the theory of lattices.

A lattice is a discrete subgroup of $\mathbb{R}^n$ that is generated by a basis of $\mathbb{R}^n$. The unimodular lattices are the ones with fundamental domains of volume 1. The even lattices are the ones where the norms of the elements are all even integers (with the usual Euclidean norm on $\mathbb{R}^n$, except in this field it's conventional to use the word norm to refer to the length squared).

Given a lattice L, write $a_n$ for the number of vectors in the lattice of norm $n$. We can form a generating function $\Theta_L(q)=\sum_n a_nq^n$.

Now comes the surprise: if $L$ is an even unimodular lattice, then $\Theta_L(\exp(\pi i \tau))$, as a function of $\tau$, is a modular form of weight $n/2$. As mentioned in another post, spaces of modular forms of given weight are finite dimensional. So we now have a vast amount of information at our disposal on the possible norms of lattice vectors.

This gives some beautiful relationships between well known modular forms and lattices like $E_8$ and the 24-dimensional Leech lattice. It's neat how the geometrical problem of finding pockets of space into which you can squeeze lattice elements can translate into discovering modular forms. (And of course the appearance of the Leech lattice hints at Monstrous Moonshine.)

As an answer to (2): I think the easiest to state application of modular forms to a domain not obviously connected to them is in the theory of lattices.

A lattice is a discrete subgroup of $\mathbb{R}^n$ that is generated by a basis of $\mathbb{R}^n$. The unimodular lattices are the ones with fundamental domains of volume 1. The even lattices are the ones where the norms of the elements are all even integers (with the usual Euclidean norm on $\mathbb{R}^n$, except in this field it's conventional to use the word norm to refer to the length squared).

Given a lattice $L$, write $a_k$ for the number of vectors in the lattice of norm $k$. We can form a generating function $\Theta_L(q)=\sum_k a_kq^k$.

Now comes the surprise: if $L$ is an even unimodular lattice, then $\Theta_L(\exp(\pi i \tau))$, as a function of $\tau$, is a modular form of weight $n/2$. As mentioned in another post, spaces of modular forms of given weight are finite dimensional. So we now have a vast amount of information at our disposal on the possible norms of lattice vectors.

This gives some beautiful relationships between well known modular forms and lattices like $E_8$ and the 24-dimensional Leech lattice. It's neat how the geometrical problem of finding pockets of space into which you can squeeze lattice elements can translate into discovering modular forms. (And of course the appearance of the Leech lattice hints at Monstrous Moonshine.)