What are the rings $A$ that appear as quotients $B/I$ of subrings$B \subseteq F$ of fields $F$ and for $I \subseteq B$ ideals? For each $A$, give an explicit formula for a ring $B$ and a field $F$.
For any ring homomorphism $\varphi: B \to F$ for some ring $B$ and field $F$, we know that $B/\ker \varphi \cong \textrm{Im }F$. If $A$ is a domain then we may let $I = \{ 0\}$, $B = A$, and $F$ be the field of fractions of $A$.
I am wondering if you could tell me (1) if my reasoning so far is correct, and (2) how to proceed from here.
In case $A$ is an integral domain, what you did works.
In general, any commutative ring is a quotient of an integral domain. For example, for each $a\in A$ define a variable $x_a$, and let $B$ be the very large polynomial ring $B=\mathbb{Z}[\{x_a\}_{a\in A}]$. There is a clear surjective homomorphism $B\to A$, and so $A$ is a quotient of $B$. Now take $F$ to be the field of fractions of $B$.