No. Elemental Hero Bladedge does not have any effects that activate. Because his effect is continuous, there is nothing for Divine Wrath to chain to, therefore it cannot be used against him.
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.
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.
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.
Both of Morphtronic Radion's effects are continuous and never 'activate'. Because they never activate, Divine Wrath can't be used against them.
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.
No, Divine Wrath must chain to a monster effect when it activates. Buster Blader's ATK increasing effect is continuous, it never 'activates', at no point does it ever use the chain, even when it is summoned, or the ATK changes. Because it is continuous, Divine Wrath cannot be used against 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.
When Marshamallon is face-down, and it is flipped by an attack, it has an effect that activates to deal the attacking player 1000 damage. This is a trigger, and Divine Wrath can be chained to it. However if Marshmallon was flipped on a previous turn, then Divine Wrath cannot be used against his 'cannot be destroyed by battle' because it is continuous and does not activate.
Divine Wrath can be used against monster effects that activate. Marshmallon has an activated effect that deals damage, Man Eater Bug has an activated effect, as does Penguin Soldier. All can be stopped with Divine Wrath. Marshmallon also has a continuous effect that stops it being destroyed by battle, and Cyber Dragon has an inherent special summon effect. Neither of these activate, so Divine Wrath can't be used against them.
Yes, it can be used on Obelisk's activated effects (Divine Wrath does not target) and most of the time, Ra's Ignition Effect effect, but when Ra summons itself, no effect apart from its own can be used in the response chain. So if Ra uses its Trigger or Ignition Effect in response to its own summon, Divine Wrath can't be used against it.
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.