If the Tuner has an activated ability, yes, it can negate that. It cannot stop a monster 'being' a Tuner, nor affect a Synchro Summon though.
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.
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, once the monarch activates its effect, there are several things you could do. Doomcalibur Knight, Forbidden Chalice, Effect Veiler, Skill Drain, Divine Wrath, Gladiator Beast War Chariot, could all negate that effect.
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.
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.
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.
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, the monster effect is still being activated whether it is on the field or not.
Skill Drain will negate its continuous effect. It can be used as Synchro Material if Skill Drain is active. Most Monster Effect negation cards will work when it activates its Ignition Effect. Divine Wrath can stop it, as can Solemn Warning, and Royal Oppression.
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, once the monarch activates its effect, there are several things you could do. Doomcalibur Knight, Forbidden Chalice, Effect Veiler, Skill Drain, Divine Wrath, Gladiator Beast War Chariot, could all negate that effect.
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.
gladiator beast heraklinos no doubt :) you can discard a card to negate any spell or trap on either players turn and if you have a divine wrath card on the field you can protect it from card effects!
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.
No, it is continuous and does not 'activate', Divine Wrath can only be used against monster effects that do activate. It is applied immediately, the moment it declares an attack, without using the chain. That means cards like Sakuretsu Armor cannot be used against it - if it was an effect that activates, Sakuretsu Armor could be chained to 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.
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.