Other mistake: Eve 6 says that ordinary human beings have 46 chromosomes, while the Adams and the Eves have 56 chromosomes. This makes 10 additional chromosomes. However, Eve 6 also mentions that they have additional chromosomes number 4, 5, 12, 16, and 22 - only 5 additional chromosomes. (00:22:40)
Suggested correction: There is no mistake. Normal humans have 23 pairs of chromosomes: two 1s, two 2s, two 22s, and either XX for females or XY for males. 46 chromosomes in total. The Eves have 5 extra pairs of chromosomes, or 10 extra individual chromosomes. So what Eve 6 said tallies up. In reality, this many extra chromosomes would surely be lethal.
Character mistake: Scully gives a description that Eve is driving a Blue Carolla. However in the next scene, Eve is driving a Pontiac, not a Toyota. (00:28:00 - 00:28:51)

Continuity mistake: When Dr. Kendrick/Eve is talking to the two little girls and she's clearing up the mess on the table, the position of the straw changes between shots. (00:29:55)

Continuity mistake: We see Cindy and Teena at the table talking with Eve 7, drinking out of paper cups. Later on, when Mulder picks up a cup from the table, it's made of glass. (00:30:40 - 00:34:30)
Continuity mistake: When Mulder grabs the twins, one of them immediately turns around toward him, however in the next shot she turns around once again. (00:41:10)
Revealing mistake: In the shot where Mulder is jumping as to avoid being burnt by fire set by Cecil the caretaker, it is easy to see that it's a stuntman instead of Mulder. (00:38:35)
Continuity mistake: While speaking with Mulder, Scully keeps her head down when in the front shot, while she keeps her head up in the back shot. (00:06:15)
Continuity mistake: A heavy rain can be noticed in the background, yet Mulder and Scully are dry as a bone. (00:20:00)
Continuity mistake: After the part where the head Kindred woman tells Mulder and Scully "I asked you not to interfere", Mulder leads Scully away. From one camera angle, he is leading her away by the hand. In the same scene, but from a different camera angle, he has his arm around her. When we go back to the first camera angle, he is once again leading her away by the hand. (00:33:10)
Visible crew/equipment: When female Marty gets out of the car for the police officer if you look at the left side of the screen you can see the male Marty actor stood there wearing matching clothing and watching the filming. (00:36:33)
Revealing mistake: When Mulder is examining the residue on the mouth of the deceased man in The Beginning, the dead body's right eye twitches.
Visible crew/equipment: In the scene at the very beginning of the episode, when the camera is going up the stairs, you can make out a man's reflection holding a camera on the stairs.
Continuity mistake: During the Re-qualification practice shoot scene Dupre as agent Willis fires until he suffers a failure-to-eject stoppage (stovepipe), despite staring at the ejection port throughout the preceding drill he fails to recognise and clear this (as any firearm user would, especially a bank robber with the partial memories of an FBI agent/instructor), and could potentially still have rounds in the magazine. At the next camera change the slide is fully back despite no manipulation from Dupre, as he removes an empty magazine. (00:19:01)
Factual error: When Mulder gives Scully the birthday card, she says that he is two months early. Given that Scully's birthday is on 23 February, the time of the episode should be sometime near Christmas. However, through the whole episode, there is no sign of Christmas time (not a single Christmas tree, no Christmas decorations, etc). (00:19:35)
Audio problem: The first scene following the intro, when Mulder and Scully step out of the car, Scully says, "I still don't get it" but her mouth doesn't move. (00:02:30)

Continuity mistake: In the close-up shot of Mulder just before he shoots Barnett, his black ear piece disappears. (00:42:15)
Continuity mistake: In the scene at the start, when the police car passes the truck, clearly it is daylight, but right after the truck shuts down it is completely dark. (00:03:25)
Continuity mistake: When Scully changes the refill of her ball-pen, in the right shot she keeps it under the desk while in the next left shot she holds it over the desk. (00:12:20)
Continuity mistake: When Mulder walks around Scully, he passes her by, however in the next shot he is again behind her. (00:16:10)