Yes. I’ve brought an empty Nalgene on a lot of planes.
Yes sure empty bottles are okay
Yes, and when I’ve forgotten to empty mine before going through the x-ray machine, I’ve guzzled 32 ounces of water in line. Nobody seems to mind.
Yes, it’s ok! You can also bring the empty bottles through the security checkpoint and then refill them on the other side, before you get on the plane.
@emilynathon: hopefully you had an aisle seat!