Q1. Does there exist an ID R in which every non zero prime ideal of type pR is maximal ideal but R is not PID?
Q2. Does there exist an ID R in which every non zero prime ideal is maximal ideal but R is not PID?
If R is a UFD, then there does not exist such examples. But is it true if R is ID or FD?
For question 2, any Dedekind domain satisfies these conditions: they're integral domains of Krull dimension $1$. For instance, any ring of algebraic integers is a Dedekind domain, and for such rings being a UFD is equivalent to being principal.
In the particular case of quadratic integers, there are only $9$ imaginary number fields such that their ring of quadratic integers is principal. For real number fields, it is not even known whether there is an infinity of them with a principal ring of integers (it is a conjecture by Gauß).