A face-up plane card that's turned face down becomes a new object with no relation to its previous existence. In particular, it loses all counters it may have had.
Immersturm
The controller of a face-up plane card is the player designated as the "planar controller." Normally, the planar controller is whoever the active player is. However, if the current planar controller would leave the game, instead the next player in turn order that wouldn't leave the game becomes the planar controller, then the old planar controller leaves the game. The new planar controller retains that designation until they leave the game or a different player becomes the active player, whichever comes first.
Any "as [this permanent] enters" choices for the card returned by the chaos ability are made by that card's owner, not its old controller or the controller of the chaos ability.
Magic: The Gathering · ™ & © Wizards of the Coast · Illus. Raymond Swanland
Card data via Scryfall. Rarix is fan-made; not endorsed by the rights holders.
