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That's mirroring, and it's expected — an iPhone can't show the slides on a tv while showing something different on its own screen. During a session, tap the expand button in the top-right corner so the phone (and the tv) show just the slide. On an iPad with Stage Manager, there's a way to keep the notes on the iPad and put only the slides on the tv — it's covered on the "tv & airplay" page inside the app.
Open Settings from the gear icon on the topic list, and adjust how often and when there. The poster and closing slide update to match automatically.
Yes — Settings has a "start over" option that unchecks every topic you've marked covered. It asks you to confirm first, since it can't be undone.
Only on your own device — which topics you've covered and your family's schedule. Nothing is ever sent anywhere. The full details are in the privacy policy.