Yes, monsters may be 'tribute set'.
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No, a Tribute Summon follows the same position rules as a Normal Summon, so a monster you Tribute Summon is either face-up attack or Tribute Set into face-down defense.
Yes, the tributed monster does not have to be face-up at all, it can be tributed while face-down.
Yes. Face-down monsters can be tributed for tribute summons, card effects, etc.
Setting is similar, yet not identical to normal summoning. a normal summon is when you Summon a monster onto the field in face-up attack position, whereas a set is when you normal summon a monster in face-down defense position. When a monster is set, it is not yet considered a summoned monster.
No, face-down monsters are valid tributes, you are not forced to flip them first. There might be one or two cards that do require the tributes to be face up, but these will specify in the effect text.