Monsters who go into the S/T zone for any reason are then treated as Continuous Spell Cards or Equip Spell Cards depending on what it is doing. It is no longer a monster card so is unaffected by anything that specifically affects monster cards, and cannot be tributed for a tribute summon, etc. However if something required you to send an Equip Spell Card to the graveyard, you could use that monster.
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Relinquished's equipped monster is destroyed if Relinquished's ATK is lower than the foe's attacking monster's ATK. If Relinquished is in Defense position, its equipped monster is destroyed if its DEF is less than the attacking monster's ATK. Either way, Relinquished remains on the field.
It is either a monster that can function as an Equip card through its own effect, such as Union monsters, or, it is a monster forcibly equipped by another card's effect such as Relinquished's ignition effect.
Yes, the tributed monster does not have to be face-up at all, it can be tributed while face-down.
Yes, the monster obtained by the effect of Creature Swapcan be tributed.
Relinquished is Ritual Summoned with the Ritual Spell Card Black Illusion Ritual. It requires a monster as a Tribute to Special Summon Relinquished from your Hand. Once Relinquished is successfully Ritual Summoned, if it is sent to the Graveyard, it is eligible to be Special Summoned from it. You cannot Special Summon Relinquished from the Graveyard unless it was successfully Ritual Summoned first.