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.
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Relinquished can only ever initially reach the field by being summoned from hand, using an appropriate Ritual Spell Card - or by any card that specifically lets you summon a Ritual Monster from other zones.Relinquished cannot be special summoned from deck by cards like Reasoning, or Mystic Tomato, etc.
No, it can only be special summoned initially from your hand by removing one earth monster from play. Effects that special summon monsters from deck, cannot special summon a monster which is 'special summon only' because these monsters have to follow their own summon method. So you cannot special summon Relinquished using Mystic Tomato, it has to be Ritual Summoned or otherwise summoned by a card that says it can summon Ritual Monsters. Likewise, Gigantes can't be special summoned from deck by an external effect, and of course can't use its own effect while it is still in the deck.
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.
No, a Synchro Summon is a kind of Special Summon.
Thousand-Eyes Idol is a Normal monster with ATK/0 DEF/0. You can fuse it with Relinquished to Fusion Summon Thousand Eyes Restrict. Thousand Eyes Restrict is banned, so Thousand-Eyes Idol is practically useless as of today.