What is $\Bbb Z^n/(a_1, \dots, a_n)$ or $\Bbb Z^n / I$ isomorphic to?

The algorithm you want is called Smith normal form. This allows to to compute the quotients as follows:

Take for example your subgroup $I=\langle (4,0,2),(2,−2,0)\rangle$. Then, we can view this as $I = A\mathbb{Z}^3$, where $$A = \left(\begin{array}{ccc} 4&0&2\\ 2&-2&0\\0&0&0\end{array}\right).$$

Apply the algorithm to put $A$ in Smith normal form and you can easily read off the quotient. This also applies to 1).

The algorithm and theorem you want to look at is the Smith normal form, which works in general for principal ideal domains.