1. Are you able to use your arcane pool to enhance a natural weapon (such as a bite)?
2. Can you use Spellstrike with natural weapons?
3. The wording of Natural Spell Combat is confusing me. Let's assume I'm only using one bite attack and no other weapons at all. I need the Natural Spell Combat arcana in order to use spell combat with my bite, correct?
With Spell Combat, you can attack with a weapon wielded in one hand. It has been ruled that a natural weapon associated with a single hand, like a claw or slam, qualifies. There's nothing in Spellstrike to suggest that the ruling wouldn't extend to that.
Without Natural Spell Combat, you can use your claw attack instead of your sword. In order to use your bite, you need NSC once, which allows you to use your bite instead of or in addition to your regular spell combat attacks.
IMO, "natural weapons" are a sloppy misnomer, they're natural attacks, not subject to "weapon"-enhancing efects.