In an analogy, Nathan says that Caleb should pretend he's "Star Trek's" Captain Kirk and "Engage (his) intellect." However, it was not Star Trek's Kirk, but rather Captain Jean Luc Picard of "Star Trek: The Next Generation" who was known for speaking the command, "Engage!"
Nathan repeats thrice the Bhagavad Gita quote "In sleep, in confusion, in the depths of shame, the good deeds a man has done before defend him". On the second and third recitement, however, he says 'defends' instead of 'defend'. As the sentence subject is the plural 'good deeds', this creates a case of subject-verb disagreement. This may, arguably, be an intentional mistake by an intoxicated character.
When Caleb is done shaving, he exits the bathroom drying his now-clean face with a towel. As he picks up his shirt, he glances over at the monitor to see Nathan conversing with Ava and then tearing up her picture. When the camera returns to Caleb, he still has shaving cream on his face.
After Caleb signs the non-disclosure agreement, he places the pen towards the right of the pad on the desk. In the next shot the pen is in the middle of the pad. From a different camera angle, when he slaps his key-card down on the desk, the pen has returned to the right of the pad.
Passing the Turing test indicates an intelligence indistinguishable from a human intelligence. Exactly what such a result implies is a matter of debate and the conclusion that the characters in the film draw - that passing would imply consciousness - is one of many accepted schools of thought on the subject.
The design of Nathan's security system is extremely unsafe: real secure environments would have battery backups for the key-cards, some doors that fail open for safety, and so on. However it's established very early in the movie that Nathan designed the security system himself, and that Nathan is a paranoid alcoholic who can't conceive of his own fallibility. Of course he designed the security system this way.
When Ava presses her drawing against the glass to show it to Caleb, in one shot it slightly slips. Apparently the contact pressure of her hand changes over time to the point where the paper starts moving. Nathan designed his android bodies to be human-like, up to and including making small errors like this; it is entirely consistent with her behavior in the rest of the movie.
Caleb should have been able to get out when he caused the power outage by inserting his key card into the computer, because he programmed the security system to unlock everything when a power outage would occur.
Caleb had already reprogrammed the security protocols when Nathan was drunk. Thus he did not need to get him drunk again on the day of Ava's escape, and Ava could have escape earlier during the penultimate power cut.
When Caleb is locked in, he tries to use his key-card to access the computer and devise a way to escape. However the key-card, rather than giving him access to the computer, shuts down the power. Either this was a part of his plan, meant to stop Nathan from escaping, or it was an amendment by Ava. It is not an error.
Although Ava is leaving behind the system by which she recharges herself, it does not follow that she will run out of power and shut down: Considering her demonstrated intelligence and problem-solving capabilities, to say nothing of her access to information, there is no reason to doubt that she would be able to find or create any number of means with which she can meet her needs before her current charge is used up.