If the monster has a continuous effect, this ceases to apply once the monster leaves the field. If Jinzo is destroyed, for example, traps can once again be activated.
If a monster has an activated effect, then chaining a destruction effect to this activation will not prevent that effect from resolving. As an example, if Chaos Sorcerer activates its Ignition Effect, or Caius the Shadow Monarch's effect is triggered, destroying them before their effects resolve, will not stop the resolution happening.
Flip Effect monsters:
These are the monsters that say "FLIP:" followed by their effect. When a face-down Flip Effect monster is attacked, it is flipped face-up at the start of the Damage Step, damage is calculated, and the monster is considered destroyed or not.
After all of that resolves, the Flip Effect monster's effect activates. After that event resolves the Flip Effect monster is destroyed and sent to the Graveyard (and the Damage Step ends).
(Note: Flip Summon is different than Flip Effects or effects triggered when flipped face up.)
After Damage Calculation effects:
Whether these effects work or not depend entirely on the type of effect. Let's say Wall of Illusion attacks and destroys a monster. Wall's effect activates after damage calculation, but since the monster was considered destroyed it does not return to the hand.
"Destroyed as a result of battle" monsters:
Monsters like Giant Rat or Masked Dragon effects still activate at the end of the Damage Step they were destroyed as a result of battle in.
"When" Effect monsters:
Cards like Peten the Dark Clown have effects that activate "when" it is sent to the Graveyard. But only if this is the last thing to happen! If anything resolves immediately after it is sent to the Graveyard you "miss the timing".
If Peten was tributed for a Tribute Summon, the last thing that happens is the monster being summoned so its effect doesn't activate.
(Confusing? Yes, but these types of cards are few and far between.)
The exceptions to the above will be any resolution that specifically needs the activated monster to remain on the field, ie, the resolution somehow refers to or affects it. Elemental Hero Prisma, for example, needs to remain on the field, for his name changing effect to apply to him.
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Destroying a monster does not flip it. But if a monster is flipped face-up when attacked, and considered destroyed by battle, then the flip effect will still activate.
The effect activates first, and you are actually chaining Trap Hole to that trigger. Trap Hole will resolve first, but it will not stop the monster effect from resolving. Unless a card says it negates a monster's effect, then it is still able to resolve its affect when flip summoned.
No, flip effect activate when the card is flipped, which will be by Flip Summon, or being attacked, or flipped by an effect. Simply being destroyed while face down, will not 'flip' the monster.
Destroying a monster in chain to the activation of its effect, will not stop the effect resolving unless it specifically and obviously requires the monster to remain face-up on the field. The lingering effect of Brotherhood of the Fire Fist - Tiger King cannot be negated simply by destroying the monster. You would have to chain a monster negation effect to its activation.
The monster (Obnoxious Celtic Guard) will flip face-up at the beginning of the Damage Step, meaning that its protective continuous effect will be active during Damage Calculation. It will therefore still be safe from monsters of 1900ATK or more, just the same as if it was attacked while face-up.