Kira tells Sisko he was having a "pagh'tem'far," or a "sacred vision." The universal translator should have interpreted that term for him, with no need for Kira to translate or repeat herself (it is unknown if Kira ever speaks English. Presumably she is always speaking Bajoran and her words are translated to English for the humans she works with).
After Sisko begins his Orb experience, there is an establishing shot of the station before the scene changes to the signing ceremony in the ward room. As the shot ends, the nose of the Defiant can just briefly be seen coming into the shot. Apparently footage of the station was reused from the Defiant's stock docking sequence and was not cut away from in time.
A visiting Starfleet officer seen wearing the "TNG" style of Starfleet uniform, with the division-colored body and black shoulders, that was put into service in 2366. That style of uniform was phased out by 2371 in favor of the "DS9/Voyager" style black jumpsuit with division-colored shoulders, and retired completely by 2373, when this story takes place, in favor of the black two-piece uniform with gray shoulders and division-colored undershirt that the rest of the Starfleet personnel wear. As the uniforms reportedly cost $3000-4000 a piece, it was likely cost-prohibitive to outfit all the extras with the new style of uniform. The Starfleet admirals are also seen with the older style of uniform, rather than an updated black-and-gray look to match the rest of the fleet, a situation which will continue into DS9's sixth season.
The Starfleet officer in the older TNG style of uniform who accompanies Admiral Whatley is not wearing any sort of rank on his collar. This is presumably Captain Rifkin, who was mentioned in dialogue earlier, so he should have 4 gold rank pips on his collar. Even if he is not Captain Rifkin, or even a commissioned officer at all, and was instead an enlisted crewman, he'd still have a rank insignia, similar to Chief O'Brien.
Sisko postulates that the markings on the spire are a coordinate system based on the position of stars in the night sky from Bajor. The painting is said to be over 20,000 years old. The positions of stars in the sky drift over time. A coordinate system that utilized star positions would not be valid after that length of time.
It is presented as though Captain Sisko is the first person to have uncovered the third side of the spire by noticing the reflection on the waterfall in the painting. The image is very apparent. It is virtually impossible that no one in the painting's 20,000 year history has noticed the reflection before.
Dax says that "Starfleet should accept a new member every week". The Federation's civilian government accepts new members, not Starfleet.
Throughout the episode Sisko wears his com badge on the gray top of his uniform instead of below the gray top, which is customary of the uniforms.
After Quark informs Sisko that it's 3 in the morning, Sisko asks Quark to save his holographic file. Sisko brought the file in himself on a data rod. There'd be no need to have Quark save it. All Sisko's progress could be saved to the original data rod.
Dr. Bashir says he can't go against the wishes of his patient and perform surgery that Captain Sisko refused. Sisko is clearly not of sound mind to be making decisions regarding his own health and his decision on the subject could easily be overridden.