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.
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no, it's a normal magic card, you can only use it when it's your turn.
As Destiny Hero - Plasma's equip effect is a "Once Per Turn" effect, yes it can, and the existing monster is destroyed.
The effect he uses to destroy a monster, is an Ignition Effect, so can only be used in your own turns, in your main phases. However his counter-adding effect will add counters in either turn.
"Relinquished" + "Thousand-Eyes Idol" As long as this card remains face-up on the field, other monsters cannot change their battle positions or attack. Select 1 monster on your opponent's side of the field, and equip it to this card (this effect can only be used once per turn, and you can only equip 1 monster at a time to this card). The ATK and DEF of this card become the same amounts as the monster equipped to this card. If this card is destroyed as a result of battle, the equipped monster is destroyed instead.
Spirit Burner is a equip spell card, it may not be used during your opponents turn. Only quick play spells (indicated by a lightning bolt) may be used then.