When is a field a nontrivial field of fractions?
Solution 1:
This is a partial answer (it answers your first question).
Every field of characteristic zero is the fraction field of some integral domain which is not a field. Indeed, let $k$ be your field and let $(X_i)$ be a transcendence basis for $k$ over $\mathbb{Q}$. Consider the ring $R$ which is the integral closure of $\mathbb{Z}[\{X_i\}]$ in $k$. Note then that $R\ne k$ (since integral extensions preserve dimension), but it's a common fact that $k=\text{Frac}(R)$.