Does this mean that "original" Cal started in the Alpha timeline because it was empty? Also to clarify when you said "repeat," does it mean: Cal starts off in alpha universe, eventually gets the souls of the 3 stuck in it, makes its way to the beta universe via bro(/dirk??), eventually dies, then repeats?

Repeat means start back at the top.  Cal doesn’t actually start anywhere, as a juju his origin is a paradox. His life cycle is a circle.