Jadzia explains the concept of an Irish wake to Worf. He attended such an event in season 5 of The Next Generation in "The Next Phase" when Ensign Ro and Lt. Commander La Forge were presumed dead.
Sisko, Bashir, and O'Brien look for Lisa's whereabouts on the planet in pouring rain yet as they stand over her body, they are fully dry.
Dr. Bashir warns Lisa that when she runs out of tri-ox, the excess carbon dioxide of the L-class planet atmosphere will cause hypoxia. However, this would actually produce hypercapnia. Rather than turning blue as described, her skin would become flushed.
The crew of the Defiant discusses how to increase their speed from warp 9 to warp 9.5 while maintaining structural integrity. Maximum warp for a Defiant-class vessel is warp 9.982. Warp 9.5 wouldn't be a problem.
The Defiant receives a distress call from an inhospitable planet that's 6 days away at maximum warp. That's dozens of light years away at least. A subspace distress beacon cannot broadcast that far, even if it was to somehow have survived the crash intact (given the state of the rest of Captain Cusak's equipment, that is a bit unlikely in and of itself).
The Olympia was returning from an 8 year exploratory mission when it crashed. The crash took place 3 years prior to the Defiant's arrival. The main body of the episode takes place in 2374. That means the Olympia left Federation space 11 years earlier, in 2363. When the Defiant crew finds Captain Cusak's body, she is wearing the style of Starfleet uniform that went into service in 2366, 3 years after the Olympia departed for its mission.
When Quark muses about how humans in the past were romantic, he pronounces "human" in the normal method, instead of "hew-mon" as Ferengi traditionally do.
Defiant's crew would have checked their computer for information about Cusak's ship immediately upon establishing radio contact and discovered the time shift.
Even on an exploration to the Beta Quardant, the Olympia would still be in contact with Starfleet and, had it not been destroyed years prior, would have known about the war with the Dominion. The fact that Captain Cusak was totally ignorant of the war should have been a major clue to the Defiant crew that something was amiss.
The Defiant crew holds a funeral service for Captain Cusak on DS9. No one but a handful of their crew, mostly the command crew, attend. Where are her other friends and family members who've been missing her for 3 years?
Captain Cusak and Chief O'Brien discuss ship's counselors, with her eventually encouraging him to speak to one. He seems agreeable to her advice, but fails to mention that neither the Defiant (an escort ship with minimal amenities) nor DS9 has a counselor. They wouldn't have one until season 7 when Ezri Dax takes on that role.
Quark tells Odo that his one-month anniversary with Kira is "this Saturday." Quark is a Ferengi, and the station runs on the Bajoran calendar. Neither the Ferengi nor the Bajoran calendar has a "Saturday." That is an Earth term.
Captain Yates refers to the Defiant as "a warship" and is not corrected. The Defiant is not a warship, it is classified as an escort ship, as Starfleet does not officially allow the production of warships.