The destruction of the Odyssey may have been classified, but if you recall DS9 at the beginning of season 3, people were leaving DS9 because it was so dangerous. If civilians knew about it, then a starship that stopped at DS9 right before being sent to the DQ would know that the Dominion exists and it is dangerous.
Not necessarily. The civilians who left DS9 were people who lived on the space station, so they were most likely aware of the Dominion because of the events of both "The Jem'Hadar" and "The Search." Sisko was concerned about possibility of a Dominion attack on the station if a fleet tried to enter the Alpha Quadrant and almost certainly would have offered civilians the chance to leave the station for their own safety.
was nowhere near DS9 until she was trying to track down the Maquis. Janeway probably read Sisko's reports on the region to gain insight into the Maquis and their conflict with the Cardassians. That Janeway had experience with the Cardassians herself would have helped matters, but there's no way to know how up-to-date her knowledge was. It's not clear if Janeway and Sisko ever spoke to each other because the latter doesn't appear in "Caretaker." He might have told her about the Dominion, but we can't know for sure.