In the flashback, Superman is wearing his modified costume when he is first possessed by Starro, which indicates that this event takes place after Justice League and Justice League Unlimited. In a roundtable on the episode's special feature, Will Friedle and Bruce Timm debated this point: Timm insisted that Superman was wearing his old costume, which raised the distressing possibility that he was possessed during some of his years with Justice League, while Friedle correctly insisted otherwise.
When fleeing Superman, Terry asks Bruce what the top speed of the Batwing is. Bruce replies, "Mach 3." Terry then asks if that is faster than a speeding bullet. In fact, it is: Mach 3 is around 2,300 miles per hour, while a speeding bullet travels a little over 1,800 miles per hour.
The flashback sequence that Aquagirl refers to are the events of the Superman: The Animated Series episode "The Main Man" in which Superman and Lobo were kidnapped by the Preserver, and after which Superman adopted the collection of alien creatures. Also, the serpent that Batman stumbles upon earlier is the same that Superman and Lobo fought together in that episode. Aquagirl also indicates that Starro witness Superman's powers firsthand. This either occurred in the Superman: The Animated Series episode "The Main Man" when Superman was fighting the Preserver, or in the Justice League Unlimited episode "For the Man Who Has Everything" when Superman was fighting Mongul, or even a combination of both scenarios.
This episode indicates that the Fortress of Solitude's location is a secret kept even from the other JLU members but Batman (Terry McGinnis) reveals that his predecessor knows its location. The Justice League Unlimited episode "For the Man Who Has Everything" confirms this fact when Batman (Bruce Wayne) and Wonder Woman visit the location to celebrate Superman's birthday.
Big Barda and Superman tell Batman (Terry McGinnis) that his predecessor was never more than a part-time member. Justice League and Justice League Unlimited stay consistent with this fact with Batman (Bruce Wayne) being a part-time member of the Justice League.