The key difference between Backing and Influence is rank. The Exalted corebook clearly states that if you have Backing, then you have rank, and that Influence does not cover rank. Backing also, if you include the Sidereal bits of it, also lets you temporarily obtain Backgrounds that you don't already have.
Maybe. Even if Influence is the same as Reputation, you don't have either. To me, Reputation is about your name being out there, people knowing who you are, getting bonuses to Social-type rolls when people are aware of your Reputation. Influence is about...well, being able to push people and get them to do stuff for you. To me, Reputation vs Influence is the same as Charisma vs Manipulation - one is getting people to do stuff because they know you and like you, and the other is because you have some sort of way of compelling them.