When a stream is offline, the api returns an empty array, meaning that a broadcaster can change the title as much as they like but it makes no difference at all to the API as there’s no stream object to be changed, it’s still just an empty array. Because the underlying API for the webhook didn’t change, no notifications will be sent out if a topic change happens while offline.
Yes, the started_at field is when the stream went live. Under ideal conditions it’ll remain constant until the broadcaster ends their stream, but if there is a short connection drop it’s possible for it to be counted as a new stream when they’re back online and there’ll be a new started_at and id.