Yes. Once Red-Eyes Darkness Metal Dragon is successfully summoned, there is a summon response window for cards like Bottomless Trap Hole. Only Spell Speed 2 or 3 effects can be used here.
You can therefore use Fiendish Chain against it, which will resolve if not negated. That means that when Red-Eyes Darkness Metal Dragon is able to activate its Ignition Effect, it will be negated when it tries to resolve.
However it is worth noting that you could always wait until Red-Eyes Darkness Metal Dragon tries to activate the effect, and chain Fiendish Chain to it, it will resolve first and therefore negate the dragon's effect too. So hitting it before it can activate is not actually necessary.
No. The text has been changed but the effect remains the same, first it was a 'cannot be special summoned except by..' monster, which means only the listed method could ever be used, it was not the kind that could be brought back assuming you summoned it properly first. Now it says "Must be summoned by....cannot be summoned in any other way", again, it doesn't matter if you properly summoned it, it can't be summoned back from the graveyard.
The turn player has Priority, so can activate Chaos Sorcerer's effect as chain link 1. Fiendish Chain can be chained to this. Fiendish Chain will resolve first, so Chaos Sorcerer's effect will be negated when it tries to resolve.
If Koa'ki Meiru Maximus was properly special summoned first by the listed text, before it went to the graveyard, then it can be resummoned by effects like Call of the Haunted or Red-Eyes Darkness Metal Dragon. If it wasn't properly summoned though, then it can't be resummoned.
Yes, Battle Fader's effect is a trigger, activated in hand when an opponent's monster declares an attack. Light and Darkness Dragon will chain to and negate this, meaning Battle Fader is not summoned and the battle phase does not end. Note that Light and Darkness Dragon does not destroy the cards it negates, so Battle Fader will remain in hand. It can be activated again when the next attack is declared.
No. Darkblaze Dragon's ATK/DEF boosting effect is a Triggered effect, triggered when Special Summoned. If it is special summoned face-down, it will be unable to trigger, and will not gain the bonus ATK/DEF.
Yes. Fiendish Chain would work on Azure-Eyes Silver Dragon on it's summoning effect. The moment Azure-Eyes is summoned you can chain link Fiendish Chain to Azure-Eyes's effect creating a chain link 2. It would resolve as chain link 2: Fiendish Chain, then chain link 1 Azure-Eyes effect.
When the Monarch is tribute summoned, it has a triggered effect that activates. Fiendish Chain can be chained to this. Fiendish Chain will resolve first, meaning the Monarch effect will be negated when it tries to resolve.
Light and Darkness Dragon's first line of text says that it cannot be special summoned. Therefore when it is destroyed and its last effect triggers, it cannot choose itself.
No. The text has been changed but the effect remains the same, first it was a 'cannot be special summoned except by..' monster, which means only the listed method could ever be used, it was not the kind that could be brought back assuming you summoned it properly first. Now it says "Must be summoned by....cannot be summoned in any other way", again, it doesn't matter if you properly summoned it, it can't be summoned back from the graveyard.
The turn player has Priority, so can activate Chaos Sorcerer's effect as chain link 1. Fiendish Chain can be chained to this. Fiendish Chain will resolve first, so Chaos Sorcerer's effect will be negated when it tries to resolve.
If Koa'ki Meiru Maximus was properly special summoned first by the listed text, before it went to the graveyard, then it can be resummoned by effects like Call of the Haunted or Red-Eyes Darkness Metal Dragon. If it wasn't properly summoned though, then it can't be resummoned.
Red-Eyes Darkness Metal Dragon's own summon does not use the chain. Because nothing is activating, there is nothing for Divine Wrath to chain to - Divine Wrath negates 'activations' of monster effects. Red-Eyes Darkness Metal Dragon's effect to special summon other dragons though is an Ignition Effect, which is a form of activated effect. This can be negated using Divine Wrath.
Yes, Battle Fader's effect is a trigger, activated in hand when an opponent's monster declares an attack. Light and Darkness Dragon will chain to and negate this, meaning Battle Fader is not summoned and the battle phase does not end. Note that Light and Darkness Dragon does not destroy the cards it negates, so Battle Fader will remain in hand. It can be activated again when the next attack is declared.
No you cannot. When a monster says it 'cannot be special summoned except by..' it means that's the only way it can ever reach the field. Five-Headed Dragon can't be special summoned, except by Fusion Summoning it. Even if you fusion summon it and it is destroyed, it can't be brought back to the field by cards like Call of the Haunted, or Red-Eyes Darkness Metal Dragon.
Montage Dragon does not have any activated effects so it has nothing that can be chained to. However if Fiendish Chain is used on it, then it will not be able to attack, and its continuous effect that grants it an ATK value will be negated, and it will be treated as 0.
No. Darkblaze Dragon's ATK/DEF boosting effect is a Triggered effect, triggered when Special Summoned. If it is special summoned face-down, it will be unable to trigger, and will not gain the bonus ATK/DEF.
Yes. Effect Veiler can chain to their activations in the main phase, and Fiendish Chain can chain to them at any time outside the damage step (meaning if they are attacked and flipped, it can't negate them).