The effect "During each of your Standby Phases: Inflict 500 damage to your opponent. That monster must be on the field to activate and to resolve this effect." starts a Chain Link. (It does not target. It is activated once during each of your Standby Phases.)
Previously Official Rulings
These TCG rulings were issued by Upper Deck Entertainment and have since been deemed unofficial by Konami. They were previously considered official and can still be correct, as long as they follow the official gameplay rules, or unless Konami has issued a ruling that says otherwise. References: 
"Nightmare Wheel" is destroyed when the targeted monster is destroyed, returned to the hand, removed from play, send to the Graveyard, etc.
"Nightmare Wheel" is NOT destroyed if the monster is flipped face-down. "Nightmare Wheel" continues to target the monster, inflict damage, and prevent the monster from attacking or changing battle position.
The effect of "Nightmare Wheel" will be negated by "Royal Decree", so you can attack or change the battle position of the monster. Also, if "Royal Decree" is active, and the monster is removed from the field, "Nightmare Wheel" remains on the field meaninglessly because its text that would destroy it is negated. If "Royal Decree" is later destroyed, the effects of "Nightmare Wheel" are not applied (such as the 500 damage) if there is no monster being targeted by "Nightmare Wheel".
The effect of "Nightmare Wheel" is still applied if the targeted monster changes sides.
Mentions in Other Rulings
Dark Spirit of the Silent: If a monster that has already attacked, or a monster targeted with the effect of..."Nightmare Wheel" is targeted with the effect of "Dark Spirit of the Silent" that monster still attacks.
Heart of Clear Water: A monster equipped with this card cannot be destroyed by a card that targets, but it may still be declared a target. So the opponent may activate "Nightmare Wheel"...targeting your monster equipped with "Heart of Clear Water".
Shifting Shadows: If you control a monster that is targeted by your opponent's "Nightmare Wheel", and you flip it face-down with "Book of Moon", and control at least 1 other face-down Defense Position monster, you can activate the effect of "Shifting Shadows". However, afterwards you must tell your opponent which monster is targeted by "Nightmare Wheel".
The Dragon's Bead: You cannot activate the effect of "The Dragon's Bead" to negate a Trap Card that has already resolved, such as a "Nightmare Wheel" the opponent activated on a previous turn.