If my LLC receives a payment as part of an insurance claim, does that count as income? Can I depreciate the item being replaced by the insurance money?

You would treat it the same as if you had sold the vehicle for $9,000.  If the vehicle was fully depreciated, you would have a taxable gain of $9,000 on the disposition of the vehicle.  If the vehicle had an adjusted basis of $10,000, you would have a $1,000 loss.  If you then buy a new vehicle for $9,000 you would treat it the same as any other business purchase.