2nd, there is magic. All characters could conceivably be born with a map engraved into their brain. Sounds like a simple illusion spell, if you really need an explanation of why the HUD is in a game for immersion factors.
Next we'll be asking for completely different storylines so we don't have to explain how we know exactly what'll happen when on our additional playthroughs.
1st. You start as a prisoner as always. And obviously a prisoner don't have any possessions with him.
2nd. You could say there is magic for everything, but I'd say it's a long shot and the least preferebly explanation. Skyrim is after all low-fantasy, and having a map engraved into their brain doesn't sound very fitting.
A map is something factual. Something concrete. Even if you know where things are by experience, you'd need a map for the player to see them. So despite your theory, buying a map is a really good and "immersive" way of introducing the map-system in an RPG like Skyrim.