No, a Synchro Summon is a kind of Special Summon.
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Yes, a Special Summon of any kind does not count towards your limit of one Normal Summon or Set per turn. You can Special Summon as many times a s you want per turn.
No tribute summon is a normal summon. It's called tribute because you have to pay the cost of the summon by sending 1 (for LV5 or LV6) or 2 (LV7+) monsters from the field to the graveyard.So basically it's same as a normal summon for a lower level monster.
Gemini Monsters are normal or special summoned as normal, but when they arrive on the field, they are considered Normal Monsters and have no effects. The next time you are allowed to Normal Summon, you can 'second summon' them, ie, they count as being summoned again, even though they never leave the field. This turns them into effect monsters and switches on their effect. Note that if you are trying to 'second summon' a Lv5 or higher monster, no tribute is needed.
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.