Lightning Vortex is non-targetting so it never really activated 'on' a specific monster, even if it's alone.
If your opponent uses Lightning Vortex and you have an XYZ monster, that monster will be destroyed, as per usual, unless that particular XYZ can save itself from destruction with an effect.
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As an Ignition Effect monster, Relinquished can only take a monster in your own turns, in your own main phases. Monsters who can activate in the opponent's turn will usually specifically state so, or be a trigger which activates when a certain condition is met, regardless of turn.
only when a dragon card is active on the playing field.
No part of Call of the Haunted or Royal Decree affect a monster's ability to attack. The Sangan will be able to attack as normal.
Well if you have another Six Samurai monster on the field it`s effect activates which allows it to destroy the monster with it`s effect but it will be destroyed unless you activate it`s other special ability to destroy another Six Samurai monster instead
There is no special summoned monster. Phoenix Gearfried's effect is a trigger which activates after the resolution of the Spell Card, but in this case since Dark Hole is destroying him, by that time he'll be in the graveyard and won't be able to activate. Let's say though that he survives destruction, perhaps he had a Card Guard token on him. If that happens, then he will trigger after Dark Hole resolves, summoning the monster. Since Dark Hole has already resolved, the newly summoned monster is in no danger from it.