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.