[Regarding the effect "If you control "King's Sarcophagus", you can Special Summon this card (from your GY)."[1]
It is not classified as either an Ignition, Trigger, Quick, or Continuous Effect.
[Regarding the effect "You can send 2 cards from your hand to the GY, including this card; add 1 "King's Sarcophagus" from your Deck to your hand, then you can draw 1 card."[1]
It is an Ignition Effect that can be activated in the hand.
When resolving this effect, add 1 "King's Sarcophagus" from your Deck to your hand. Then, if you successfully added it to your hand, you can draw 1 card.
Adding "King's Sarcophagus" from your Deck to your hand and drawing a card do not occur simultaneously.
[Regarding the effect "If another card(s) you control leaves the field by an opponent's card effect while you control this monster (except during the Damage Step): You can send 1 card on the field to the GY."[1]
It is a Trigger Effect that can be activated in the Monster Zone.