Kai and natural English

I've been thinking about natural English, storytelling and translation. When I attended formal schooling I was taught that sentences starting with 'and' are not proper English. I don't think my teacher went so far as to say such sentences were ungrammatical, but just not proper.

One of the goals of a good translation is that it achieves a certain naturalness. The translation doesn't sound like the language it came from but the language into which it was translated. When reading the Christmas story in some new English Bible translations (not The Message) I was surprised to see that several of the sentences started with 'and' or 'but'. This is very common for the source language, Koine Greek - 'kai'. But how common is this for English? Does starting sentences or paragraphs with 'and' or 'but' mirror spoken English? Is the rule prohibiting using 'and' sentence initial specifically for written English? Is it a reflection of the difference between spoken and written language, or possibly a reflection of a regional variety of spoken English?

I thought I'd start listening to people telling their stories and how they begin major sections. Well, just yesterday was a good time to listen as we had a 2 hour meeting and the presentations were interspersed with stories. One presenter told a story, where after the introduction each section started with 'and' and the last section started with 'so'. It was in the last section where the significant action happened, marked by language like 'he was stunned'. The transition in these paragraph starters marked a significant change in the story to me as the listener.

What do you think?