When “Neko Mane King” ends your opponent’s turn, your opponent proceeds immediately to his/her End Phase, even if “Neko Mane King” is sent to the Graveyard during the Damage Step.
“Neko Mane King” doesn’t actually end the turn, it advances play to the End Phase, so “Destiny Board” can still be used, etc.
“Neko Mane King”’s effect activates even if it is sent to the Graveyard while face-down on the field, or sent to the Graveyard from your Deck or hand.
When “Neko Mane King” is sent to the Graveyard by the effect of “Delinquent Duo” or “Card Destruction”, “Neko Mane King”’s effect is activated after those cards completely resolve all of their effects.
“Neko Mane King”’s effect only activates when the owner controls it; if the opponent controls “Neko Mane King” then its effect is not activated.
When “Neko Mane King” is designated by “Prohibition”, its effect is not activated.
Netrep Rulings:
“Neko Mane King’s” effect resolves immediately after the chain that destroyed or sent “Neko Mane King” to the Graveyard resolves.
“Neko Mane King’s” effect does not activate if it is destroyed as the result of battle.
“Neko Mane King’s” effect moves the current Turn to the End Phase.
If a player chains “Call of the Haunted” to “Mystical Space Typhoon” to Special Summon “Neko Mane King,” the opponent’s turn does not end as it is not the opponent’s effect that destroys “Neko Mane King.”
If “Neko Mane King” is in a player’s hand, the opponent’s turn will end if the opponent plays “Card Destruction” after the resolution of “Card Destruction.”