If I kill Andy the robot before I leave the vault, can I save Beatrice from getting killed by him for having a sprained toe? (Trouble at the Homefront)

I don't think so.

Why save Beatrice anyway? I think it's rather comical to have the ignorant vault dwellers get a taste of reality.

Just liked to know if it was possible. But I don't intend on starting yet another game, only to find out.

I believe that prior to completing the Escape! quest Andy is considered essential, so you can't kill him prior to the Trouble on the Homefront quest.