Is it safe to gently throw my baby in the air and catch her?

No. It is not safe to throw a child in the air and catch them. The vertebral blood vessels are fragile and when they come down as you catch them it causes a small snap in the neck that can tear these vessels. It is fine to lift them in the air to give that buoyant sensation as long as you do not release the child. Be aware of any head snapping back.
No. Dr.Franz has a nice basic explanation. The focus of safety is avoiding situations where unexpected events result in tissue damage. You may not plan to fall, trip or otherwise be unable to catch this bundle. So even if you didn't hurt baby with the toss up, a "humpty dumpty" event may happen on the way down. Save the horseplay for your school age kids.