You can tribute a special summoned monster, yes.
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Yes, there is nothing inherently within the tributing or summoning rules which would stop you from tributing a monster you summoned the same turn you summoned it. Obviously, due to the regular summon rules, you can not Tribute Summon the monster you just Normal Summoned on the same turn; it would count as two Normal Summons. But if the first summon was a Special Summon, you had used "Double Summon", or wanted to tribute it for an effect like "Enemy Controller", then it's totally fine to tribute that monster for them.
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.
um, yeah... all monster can be normal summoned, flip summoned, special summoned etc unless specified otherwise. eg, ancient gear golem says "this card cannot be special summoned".
Yes, you can attack with the Tribute Summoned monster unless a card effect tells it not to or it is after the Battle Phase.
yes