What's the context in the "Hail" part of Hail Mary? I was just thinking, that's a strange greeting. Didn't people back then usually say "hail" in the context of "Hail Caesar"?  And apparently Mary thought that was a strange greeting too, even stranger than the fact that it was an Angel who greeted her thus. 

So, according to the Catholic Church, does Hail (or Ave) in the sense that Gabriel uses it in Luke, actually have anything to do with addressing royalty?   Has it ever been posited that the Archangel Gabriel would use that terminology only in reference to the Holy Mother of God?
The "Hail" would be a normal greeting. I've encountered some people who believe that when Catholics say "Hail Mary.." they're worshiping her. To my understanding the original meaning doesn't necessarily connote addressing royalty, and especially not worship. Those contexts to the word might be a much more modern development.

Even in today's English we see Hail used with regular contexts instance; if someone "Hails a cab", are they worshipping the car or the driver? No; they're flagging it down.

In movies, especially in naval ( or space sci-fi contexts), characters sometimes say "We're being hailed by the USS Magellan" or some such. Again, obviously nobody is honoring or worhsipping the crew or the ship- they're greeting it.