The word then is a common and useful word for connecting ideas in a sentence. Unfortunately, problems can arise from not understanding exactly what role — or what part of speech — then plays in a given situation. There are four options for then, though some writers try to shoehorn it into a fifth option.
Sometimes, then is a noun meaning “that time” — one hopes that the exact time is evident from context. For example: “John contracted a computer virus when he was twelve. Since then, he's only been able to count in hexadecimal.” This type of usage rarely presents problems for writers, editors, or readers.
Other times, then is an adjective meaning “former, belonging to a mentioned time” and is paired with a noun to create a phrasal adjective. Like other phrasal adjectives, the norm is to hyphenate it if it comes before what it describes but leave it open if it comes after, as in this fictitious example: “In 1864, then-governor of Ohio John Brough sent fruitcake to everyone on his Christmas list. Oliver Morton, then governor of Indiana, returned his fruitcake with a note bearing a single word: 'Humbug.'” I think this usage sounds a little awkward to most copy editors, and so they reword it, but there's certainly nothing wrong or difficult about it.
The majority of the time, though, then is an adverb.
In my experience, the most common problem with then, apart from confusing it with than, is writers trying to use it as a coordinating (copulative or adversative) conjunction, thus creating a comma splice.
INCORRECT: Jeremy went down on one knee, finally ready to propose to Eileen, then he passed out.
This can be corrected in three ways:
REPLACE THE COMMA WITH A SEMICOLON: Jeremy went down on one knee, finally ready to propose to Eileen; then he passed out.
INSERT A CONJUNCTION AFTER THE COMMA: Jeremy went down on one knee, finally ready to propose to Eileen, but then he passed out.
REPLACE THE COMMA WITH A PERIOD: Jeremy went down on one knee, finally ready to propose to Eileen. Then he passed out.
The last solution presents another question. When then is used as a sentence-modifying adverb, should it be followed by a comma? I think that generally the answer is yes. When giving the comma consideration, it may help to mentally replace then with another adverb that serves the same (temporal) purpose — later or now, for example.
But there's more to good writing than grammar. Good copy editors consider issues of rhythm and mood. The pauses commas add slow a sentence, and sometimes speed is called for. If the rhythm of a particular paragraph calls for a short, punchy sentence — especially if it's a bonafide punch line — the comma may only get in the way of that rhythm.
Ultimately, exactly how you fix a misused then is an editorial decision based as much on the needs of the work itself as on any grammatical guidelines.
Further muddying the water is the fact that then is also part of the correlative conjunction pair if…then. So, yes, then can be used as a conjunction, but never on its own. The salient point to remember with if…then, and with all correlative conjunction pairs, is this: “Correlative conjunctions must frame structurally identical or matching sentence parts…; in other words, each member of the pair should immediately precede the same part of speech.” (CMOS16, 5.195)
So keep your eyes on this little word, because it does get misused, every now and then.