In trying to answer my problem (1) above - I cannot see why the ideal is a prime ideal - I was looking at definitions of prime ideals and trying to reason from there.
I just looked up the definition of a prime element in D&F to find the following on page 284:
The non-zero element is called prime if the ideal (p) generated by p is a prime ideal!
So the answer to my question seems obvious:
prime prime ideal
Although this now seems obvious, I would like someone to confirm my reasoning (which as I said now seems blindingly obvious! :-)