In the usual definition, a principal ideal domain $R$ is also assumed to be an integral domain. However, the property that every ideal is generated by a single element does not seem to immediately imply that the ring is integral. Is this correct and if so:
Do there exist rings where every ideal is generated by a single element and has zero divisors?
I am most interested in the case where $R$ is commutative with unity, but don't mind examples where these properties don't hold.
Also, assuming there are examples, is there any reason why we make this assumption?
Yes. Such rings are called principal ideal rings. An example of such a ring would be $K[x]/(x^2)$, where $K$ is any field.
In fact, a theorem of Hungerford states that any principal ideal ring is the direct product of quotients of principal ideal domains.