The puck has to be grabbed and dragged. Tapping anywhere else does nothing, on purpose: a hand that brushes the screen in the dark should not be able to sweep the sound from one end to the other.
That is deliberate. The app is meant to be used by eyes that are adjusting to sleep, and a bright panel at midnight is not a cosmetic problem — it suppresses melatonin. The puck is the one thing allowed to glow, so a hand can find it in the dark.
From the lock screen or Control Centre, like any other audio. The AirPods button works too.
It is the sleep timer, closing slowly. There is no clock face on purpose — a number to look at is a number a sleepless person will look at. Press and hold the timer row if you want the figure anyway.
By design. Moving to the phone's own speaker at three in the morning would wake whoever is lying next to you.
Yes — choose NIGHT, which is eight hours followed by a three-minute fade. Not "forever": a phone that plays until the battery dies is a phone that plays into the morning. The long timers — one hour and up — are part of a one-time unlock; the sound itself, the whole surface, and timers up to thirty minutes are free for ever.
Sound Terrain cannot silence your phone — no app is allowed to. Turn on Sleep Focus in iOS if you want an undisturbed night. If Siri reads your messages aloud, that is Settings → Notifications → Announce Notifications.
On your device, and nowhere else. There is no account and no cloud sync. Deleting the app deletes them.
Anything else — a bug, a sound that does not behave, an idea: