I can find ample evidence that MongoDBs are always kept in the order given when a document is inserted or updated, but I just can't find it explicitly spelled out in the documentation for MongoDB.
So I'll go ahead and ask the [probably stupid] question: are arrays in MongoDB documents kept in the order specified when they were inserted/updated?
And sharding/replication/etc. do not effect this ordering, correct?
If you are speaking about the order of an array value of a document field, then yes the order is always kept. See this question.
If you speak about the order of documents in a collection, to my knowledge you shouldn't count on any order. If you need an order, you should implement your own