When this was discussed a few years ago, I believe the conclusion was that yes, the battery would gain weight as it depleted in energy.

No, but we don't walk around through a sea of graphite. I would expect that a properly filtered air intake would be likely, since a chunk of lithium (a solid at room temperatures, and this quite dense) would need vast volumes of air to react with.

You are right, I completely misunderstood how Lithium-air batteries work.