1: no, you are not using your Bite attack.
2: no, you are not using your Slam attack.
3: no, you are not using your Claw attack.
4: no, you are not using your Claw attack.
5: no, you are not using your Unarmed Attack.
6: Natural weapons and manufactured weapons (designated as such for ease of comparison) behave differently than each other. Manufactured weapons are in this game defined as those that gain extra attacks based on a high Base Attack Bonus. Natural weapons are made separately from your BAB. This allows you to make a full attack with a sword, and then add a Bite attack at full BAB with a -5 penalty for it being your secondary attack. In the case of, say, an animal, we'll go with a bear, it will make a primary natural attack at full BAB, and then any secondaries at -5. It does not matter the BAB of the bear, it gets to make these attacks unless they are otherwise obstructed from doing so (only standard action attack, holding an object preventing a claw, etc.). Unarmed strikes are, for the purposes of this game, manufactured weapons, not natural weapons. They do share some similarities and benefits of natural weapons, but they do not ignore BAB, so they are limited by your BAB and incur TWF penalties as normal if you add them to an attacking routine with, say, a sword. Your UA is made with any part of your body, but if you attempt to make an UA with a claw, then it's still an UA, and not a Claw attack, so it's still Bludgeoning (barring an effect that changes damage type, such as Versatile Strike) and you don't get any effect that triggers on a Claw attack (such as Rend or Improved Grab).
EDIT: take those links with a grain of salt. They are not the final say of rules. They are around the fifth line I believe.