Prime Material Dragon has an activated effect that can chain to and negate destruction effects. If it ever uses this effect, Divine Wrath can be chained to it, and Prime Material Dragon will be negated and destroyed, meaning the original destruction effect resolves correctly.
Note that as a Quick Effect, Prime Material Dragon is unable to save itself from spell speed 3 destruction cards like Divine Wrath. Note also that Prime Material's effect to change direct damage into lifegain, is continuous and Divine Wrath can't be used against it.
If you mean Dark Valkyria's 'second summon' to activate its Gemini effect, then no. This is simply another summon, exactly if it was done from hand, it just happens to remain on the field throughout. It is not an activated monster effect, and can't be stopped with Divine Wrath.
However you can use Divine Wrath on either of its two Ignition Effects.
Yes, "Obelisk's" effect can be negate by "Divine Wrath". "Divine Wrath" activates in response to "Obelisk's" effect but does not target the card, it only negates it then destroys it.
Blue-Eyes Shining Dragon cannot stop Divine Wrath, because Divine Wrath does not target it, nor can BESD chain his Quick effect to a Counter Trap.
Divine Wrath can only be used against activated effects - Obelisk has two activated effects, his destroy opposing monsters one, and the trigger that sends him to the graveyard on the turn he was special summoned. Divine Wrath can be used against either of these - note Divine Wrath does not target the monster, so Obelisk's protection effect won't stop it.
All of his effects are Triggers, which are activated effects and use the chain. You can chain Divine Wrath to any of them, to negate and destroy him.
Arcana Force 0 - The Fool's second effect only concerns effects that target him. It is continuous, so, Skill Drain will both negate The Fool's 'cannot be destroyed by battle' effect, and his own negation effect. He can be destroyed by battle now, and the opponent can target him without their effects being negated. Divine Wrath however has to chain to activated effects. Since both of these effects are continuous, Divine Wrath can do nothing at all.
No, you can use Divine Wrath on your own activated monster effects if you wish. Just as long as it is indeed an effect that activates and therefore uses the chain.
Yes, you can do this. When Divine Wrath resolves and tries to destroy Thought Ruler Archfiend, you can pay 500 LP and destroy Telekinetic Shocker's effect instead.
Mirage Dragon's effect is continuous and can't be chained to. No activation means no Divine Wrath, Divine Wrath can only be used against activated monster effects.
Yes, "Obelisk's" effect can be negate by "Divine Wrath". "Divine Wrath" activates in response to "Obelisk's" effect but does not target the card, it only negates it then destroys it.
Blue-Eyes Shining Dragon cannot stop Divine Wrath, because Divine Wrath does not target it, nor can BESD chain his Quick effect to a Counter Trap.
Divine Wrath can only be used against activated effects - Obelisk has two activated effects, his destroy opposing monsters one, and the trigger that sends him to the graveyard on the turn he was special summoned. Divine Wrath can be used against either of these - note Divine Wrath does not target the monster, so Obelisk's protection effect won't stop it.
Divine Wrath is a counter trap. When a monster's effect activates (therefore it cannot be used against an effect that does not 'activate') you can chain it by discarding one card. When Divine Wrath resolves, the monster's activation and effect is negated, and the monster is destroyed. The monster does not have to be in hand for Divine Wrath to be used, it can stop them no matter which zone they are in, as long as it is indeed an activated effect.
All of his effects are Triggers, which are activated effects and use the chain. You can chain Divine Wrath to any of them, to negate and destroy him.
Yes. It doesn't matter what zone the monster is in, Divine Wrath can be used against it as long as it 'activates'. It doesn't matter that Divine Wrath is unable to apply a destruction effect due to the monster being in the graveyard, that's not a condition of activation.
You cannot use Divine Wrath against the initial activation because it is only a trap card at this point. However after it has resolved, it is now a trap and a monster, so if it activates its Ignition Effect, Divine Wrath can be used against it.
Arcana Force 0 - The Fool's second effect only concerns effects that target him. It is continuous, so, Skill Drain will both negate The Fool's 'cannot be destroyed by battle' effect, and his own negation effect. He can be destroyed by battle now, and the opponent can target him without their effects being negated. Divine Wrath however has to chain to activated effects. Since both of these effects are continuous, Divine Wrath can do nothing at all.
the trap card -divine wrath-