Robert Houdin, as far as I understand, is responsible for the tux and top hat image. Before him magicians wore robes and turbans.
Houdin was attempting to make magic more relevant to his audiences. At the time, an evening at the theater called for one to dress nicely. A lady would wear a nice dress, a man would wear a suit or tux. So Houdin wore a tux to appear classy.
The wand has been around for ages. I assume the black wand with white tips comes from matching the tux look, and having it look the same on both sides allows it to be manipulated in a variety of ways.
Abracadabra is ancient. Possibly derived from the god Abraxas, possibly Hebrew, maybe Aramaic, and it means something along the lines of "I create as I speak". It was used as an incantation to ward off malaria in Roman times.
https://en.wikipedia.org/wiki/Abracadabra