To publish a story or stream, all the sections in your left-hand menu need to be checked/ticked. You will then see a "Publish" snack bar pop up and you can proceed to publish your story or stream.
You can also select the options menu from the dashboard and publish from there. Go here for more details.
Changes made to a story/stream are auto-saved. You will need to re-publish to make your changes available on the app.
When you add/remove access to your story or stream in Targeting, you will not need to re-publish. The added/removed groups will automatically have access/have their access revoked.