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.
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.
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.
No, "Uria" can only be Special Summoned from your hand.
It can't be Normal Summoned or Set, and can't be special summoned except by removing from play a Stardust Dragon from your Extra Deck.
You can tribute a special summoned monster, yes.
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.
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.
Ritual monsters may only be special summoned from the graveyard if they were successfully summoned by Ritual Summon. If they were sent there in any other way - like simply discarded from hand, or improperly summoned by Ritual Foregone (it special summons but does not Ritual Summon), then you cannot revive them from the graveyard with cards like Monster Reborn or Fulfillment of the Contract.
No, "Uria" can only be Special Summoned from your hand.
It can't be Normal Summoned or Set, and can't be special summoned except by removing from play a Stardust Dragon from your Extra Deck.
No, you can't. The only time Hamon can be special summoned is while he's in your hand, by following the text on his card. If a monster can be special summoned from other zones, it will say so on the card.
No, they cannot be special summoned except by following the text on their cards, and this can only be done while they are in your hand.
No, because you cannot Special Summon Relinquished without the Ritual Spell Card "Black Illusion Ritual", since Relinquished is a Ritual Monster Card (That's why the Card is colored dark blue). But you can special Summon Relinquished from the graveyard with cards like Monster Reborn, Call of the Haunted etc., as long as it was properly Ritual Summoned with the Ritual Spell Card before. If not, you are unable to Special Summon him. Hope that was not too much information at once ;P
Relinquished can be obtained in the following sets:Dark Beginning 1Master Collection Volume 1 PromosSpell RulerStarter Deck: Pegasus
It is a spellcaster ritual monster which can be ritual summoned with the ritual spell card "black illusion ritual"
Any appropriate effect can summon an Earthbound Immortal from deck, they have no innate restriction against this. Cards like 'Reasoning' can do it, and in the case of 'Wiraqocha Rasca', that one can be summoned by 'Mystic Tomato' (though it does not get its ATK gaining effect unless Normal Summoned).
You can tribute a special summoned monster, yes.