Charlie Francis was killed earlier in the season, with a shape shifter replacing him. In episode 4 of season 2, the shape shifter is also killed. Yet in this episode, Charlie re-appears as if nothing happened.
Before organs are removed from the donor, brain death must be verified (usually by two independent doctors with no prior connection to the patient), not the ventilator being switched off. The ventilation via endotracheal tube must continue until the aorta is cross-clamped, and only then be switched off. The first organ to be recovered is always the heart due its high oxygen demand, the last one the liver, not like it was shown here.
The FBI case file of Lisa Donovan, the 17 year old girl, shows her Date of Birth as 19 November, 1982. Assuming this is a season 1 episode, and takes place either in 2008 or 2009, this would place the girl's age at 26 or 27.
Right after the credits, in the scene where Olivia is introduced to the Navy Officer, he identifies the possible treason suspect as "a Petty Officer named Andrew Rusk" but the photo is of a "Chief Petty Officer" and would never be referred to by anyone in the Navy as simply "Petty Officer" in any reference to their rank. Additionally, they stated the numbers referred to "lineal identification numbers for sailors in the U.S. Navy" but only Officers have lineal numbers, not enlisted personnel (like Chief Petty Officers).
Olivia states that according to the rigor of Rusk's body, he'd been dead for 3 days.
Rigor mortis usually lasts around 8 hours after death at room temperature. There would not be any rigor in a dead body at 3 days and therefore rigor cannot be used to estimate that time of death.
A priest asserts that the Church views exorcisms as mere superstitions these days. This is factually incorrect; the Catholic Church not only believes in the fact of possession, but employs Exorcists in most dioceses. Exorcisms are performed after consent of the relevant bishop and after an examination of the patient by doctors and psychiatrists in order to determine that the affliction is not natural in origin.