There doesn't seem to be any reason why O'Brien goes to the shuttle bay with Dr. Pulaski. He wasn't asked to go, Dr. Pulaski doesn't speak with him, he's just there for no apparent reason except as a witness to the duplicate Picard disappearing.
When Picard phasers his duplicate in the shuttle bay, the duplicate takes a hit just below the right side of the ribcage in the general vicinity of the liver. But when Dr. Pulaski examines him moments later, the scorch mark on his tunic is in the center lower abdomen.
The first time Picard and Troi enter the sick bay to check on the double, you can see right through Picard (the one entering sick bay, that is). To best see it, watch the black/lower part of Picard's uniform just as the doors slide open--the light and indents on the wall behind him are clearly visible.
Picard doesn't command the computer to override the safety setting established for on-board phasers, nor does he adjust the phaser's setting... yet he takes a phaser and kills his double with a single shot.
Picard mentions that the shuttle has no warp capabilities. However, later in the series, the shuttles are capable of warp.
At the beginning of the episode, Riker is eating with Geordi, Data, Worf, and Dr Pulaski when Picard calls Riker to the bridge. In the next shot, Riker enters the bridge and Data and Worf are with him, but they weren't called.
Riker claims to be making an "omelette" but the eggs end up looking just like scrambled eggs.
The last time Picard goes to sick bay, Troi follows him. How is it that she is in sick bay when Picard arrives?
Troi was ordered to stay with the second Picard to communicate with him, so why does she leave after the discussion with Dr. Pulaski?
Picard: Well, they say if you travel far enough, you will eventually meet yourself. Having experienced that, Number One, it's not something I would care to repeat.
Picard: So you're saying I should just sit down, shut up, and wait?
Riker: Well, I wouldn't have put it exactly like that.
Wil Wheaton is credited but doesn't appear.
The shuttle discovered adrift is the Enterprise's "Shuttlecraft 05". Although no one says it aloud, the name El-Baz is printed on both sides near the front. It's especially visible when they note the antimatter explosion damage.
Originally this episode was supposed to be a precursor to "Q Who?", where Q would have explained that he created the temporal anomaly in this episode as another of his tests. The idea was later dropped.