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.
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Yes. Kasha's effect to return all monsters to deck is a triggered effect, triggered on special summon. Fiendish Chain can be chained to this, and will resolve first. That means Kasha's trigger will be negated when it tries to resolve, and nothing will be shuffled back into deck. Kasha's ATK will remain at 0.
Flip Effects are Triggered Effects, so if a player flipped a flip effect monster, the effect will trigger as chain link 1. Fiendish Chain can be chained to it, to negate the effect when it tries to resolve. However, if the monster is flipped by an attack, then the trigger happens during the Damage Step. There are restrictions against what you are allowed to activate during the damage step, and Fiendish Chain can't be used here. So it could not stop a Flip Effect at that time.
You cannot use Royal Command to negate and monster whose effect activates when it is flip summoned, as Royal Command only activates for FLIP: monsters.
Cyber Phoenix will only negate an effect if it targets, and if it targets only 1 monster, which is one of your machine monsters. It would therefore negate Sakuretsu Armor if used against one of your machine monsters. It would not stop Icarus Attack, which targets two monsters. It doesn't matter if it only targets one of your machines, it is still invalid due to targeting two monsters in total.
No. Stardust Dragon's effect only negates and destroys a card which effect DESTROYS a card on the field, however Caius the Shadow Monarch REMOVES FROM PLAY a card on the field.