Why did the Lil Cal that came to the Beta Universe with Bro have Caliborn/Arquisprite/Gamzee in him, while the Lil Cal that came to the Alpha Universe with Dirk was empty?

Because the Cal that went with Dirk was destined to be filled and it could be filled if it was already full. Likewise, the essences in Bro’s Cal were needed for Doc Scratch’s creation so it needed to have all them naughty boys shoved up in it.