In order to activate the effect of “Mystical Knight of Jackal”, the monster that “Mystical Knight of Jackal” destroyed as a result of battle must be your opponent’s own monster, not one of your monsters that your opponent controls.
“Mystical Knight of Jackal” sends a monster to the Graveyard as a result of battle, and THEN returns it to the top of the deck, so “Sangan” or “Witch of the Black Forest”, etc., will activate their effects.
If “Banisher of the Light” is in play, the monster is removed from play and “Mystical Knight of Jackal” does not activate its effect so the monster does not return to the top of the Deck.
If “Mystical Knight of Jackal” attacks a face-down “Man-Eater Bug”, and “Man-Eater Bug” destroys “Mystical Knight of Jackal”, then the effect of “Mystical Knight of Jackal” is not applied.
“Mystical Knight of Jackal” does activate its effect even if “Necrovalley” is in play.
When “Mystical Knight of Jackal” attacks “Mystic Tomato”, during the End Step both effects activate simultaneously once “Mystic Tomato” is sent to the Graveyard, so they form a chain. The turn player’s effect is Step 1 of a chain, and the opponent’s is Step 2; so “Mystic Tomato” resolves first, and a monster is Special Summoned and the Deck shuffled. Then the turn player chooses whether to place “Mystic Tomato” on top of the owner’s Deck with the effect of “Mystical Knight of Jackal”.