If the data about the Expanse and the spheres is so valuable, it ought to be backed up in different storage (i.e. RAID) against the accidental or intentional deletion or the data corruption. Not to mention the access restricted to the specific crew personnel. The computer didn't prompt whether the user wanted the database deleted or for the password to delete it.
When the zealot's space ship is let go from the docking clamps of Enterprise, it floats off and comes to a stop. In reality it would continue its course, as there's not resisting force in space that would slow the ship down.
Archer talks to one of the extremists about his wife's visit to Dr. Phlox and about her pregnancy. So clearly Phlox discussed the matter with Archer. This is wildly inconsistent with Phlox's previous insistence on absolute doctor/patient confidentiality.
If the Pri'Nam has been reading all about Enterprise in the ships data base and the Captain's logs, he would have surely come across mention of the transporter and it's use, and therefore wouldn't have been tricked by the fake execution.
D'Jamat learns the history of Archer's mission by reading the Enterprise computer files. The files are written in English, a language with presumably no correlation to the written language of D'Jamat's people. While the Universal Translator lets someone understand unfamiliar speech, it cannot teach someone how to read.
D'Jamat and Archer have a conversation about Archer's torture of Orgoth in Anomaly (2003). The conversation is regarding killing of innocents, but Archer fails to point out that Orgoth had just attempted to pirate their ship and leave them to die, so he was not an innocent.
T'Pol grapples with one of the extremists, putting him briefly a headlock. But given where she is standing, there was no reason not to use the Vulcan nerve pinch.