How is your power state when doing this testing? Battery, charger, or charger plus battery?
I've seen similar symptoms when there was a power problem. I would recommend trying to disconnect the charger and try booting - internal and/or external drive, on battery power only, and then open it up, disconnect the battery, and try doing the same thing on charger power only.
Not on your same model Macbook Air, but I've found faulty battery on an old Macbook Pro caused symptoms similar to how you describe and diagnosed it with the above method.