Continuity mistake: When Flash plays tag with Faust ("Now you see me, now you don't"), there is a brief shot in which he rushes at Faust - but not inside Hyppolita's palace, but in some high-tech laboratory. Obviously a shot from some earlier series, since the coloring and style are different from the Justice League's.
Continuity mistake: The surveillance video of Hawkgirl talking to the Justice League is different than what we saw in the previous episode. She doesn't move towards Bruce in the video, people are in different positions, and Clark has glasses on in the video.
Other mistake: When Lex finally manages to open Batman's utility belt and spills out its contents, we see several hand grenades dropping out of the pouches. However, these grenades are standard military egg grenades, and with their size and bulk they should have bulged out the square pouches in which they had been kept - if they had ever fitted inside in the first place.
Plot hole: Hippolyta tells Hawkgirl of Aresia's background of being born in the outside world, the war in her country and right up to her drifting to the shores of Themyscira. It would be impossible for her to know anything of Aresia's background as in part two, it's explained that the captain of the ship who saved Aresia died shortly after reaching Themyscira and therefore would not have been able to tell Hippolyta anything.