Can you start a sentence with 'then'?

Of course you can, especially for narrative effect (storytelling).

Grammatically speaking, 'then' is a sentence adverbial - that is to say an adverb which describes a whole clause instead of just a word or phrase (sentence fragment functioning as one part of speech). Since it introduces the idea of 'what happens next,' it is clearly connected to whatever has been stated in the preceding sentence.

Incidentally, adverbs which express chronology (first, next, then, afterwards, finally, etc...) are called sequence adverbs since they indicate order of events.