Does Dazzling Aurora enter the field first before the When You Play Me effect of Soul Gorger resolves? Basically, does Prefect exile Aurora?

Yes — Aurora enters the field before Soulgorger's "When you play me" effect resolves, so Prefect cannot exile her.

Chain items finalize in the order they were appended, and a Gear resolves immediately upon finalizing (Rule 337.1.b, 337.2). So the order is: Soulgorger enters → Aurora resolves and enters as a Gear → then Soulgorger's WYPM trigger resolves and plays Ravenbloom Prefect from the trash. When Aurora was played, Prefect was still in the trash, and its "When an opponent plays a gear" ability is only active while Prefect is on the board — so it never triggers on this Aurora. Prefect enters after Aurora is already in play, and its trigger doesn't retroactively fire.