This is a question that interests me too. In fact, one of my tutees asked me this question two weeks ago. He said that he saw a sentence in his text starting with an "and" and that was how he explained his usage of that conjunction at the beginning of one sentence in his essay. It was hard to explain to him why it did not fit in that particular sentence. However, I think, generally using conjunctions at the beginning of a sentence makes them less of connectors. It seems like it has an emphatic value.
I wonder if we should put a comma after them in these cases.