Why is Ethan Hunt's wife allowed in the middle of IMF's headquarters at the end of the film? The organization is so secret, that they operate from the basement of a front business! That's a huge security risk they are taking for what is essentially a feelgood gesture. Imagine a real-world parallel: the wife of a high-ranking Pentagon official is kidnapped in order to get at her husband, and then she is rescued. I am quite certain the people at Pentagon wouldn't go, "Sorry we caused you so much trouble! To make up for it, come mingle with your husband's co-workers in a restricted area!"
She would need to be debriefed, and the parallel is not the same because the existence of Pentagon officials is not a state secret; the existence of the Impossible Mission Force is. Plus, in the process of the "rescue" she ends up saving her husbands life, shoots a traitor to the agency dead, and overhears sensitive information about a recently stolen weapon of mass destruction. She is knee-deep in it and merely learning about IMF gives her privileges.
Why does Davian torture Ethan, asking him where the Rabbit's Foot is? Ethan had it when he was drugged in a limo, and Davian's henchman should find and give it to his boss.
As explained by the mole after the interrogation, Davian did the ruse to make absolutely sure Ethan did not switch it out for a fake. Ethan claiming he gave Davian the real one through the count of ten thinking his wife was at gunpoint verified it for Davian.
Though realistically this is a terrible way to verify anything. Once Ethan thinks his wife's life is on the line, obviously he'll say whatever it takes to save her, whether it's true or not. This actually comes up in the interrogation; Ethan starts by telling the truth but Davian says he doesn't believe him, so Ethan desperately switches to a lie that he hopes Davian will believe ("The Rabbit's Foot is in Paris"), at which point Davian casually dismisses the lie. But how exactly does he manage that? He has no way of knowing which story is true. We're left to assume that Davian is a Living Lie Detector for targets in high-stress situations, which isn't something you can rely on in real life.
Aside from an interrogation tactic, Davian is a sadistic asshole who really, really, REALLY hates Ethan and thinks it'll be hilarious to see him squirm and beg and cry when he sees Julia take a bullet.
Why didn't Ethan give a "shoo" gesture to Davian's bodyguard in the bathroom?
Ethan believed that the bodyguard would do him the courtesy of pissing off when Davian makes it clear he's not needed. When the bodyguard still approaches him, Ethan does try to shoo him away until he gets Davian's voice.
Why didn't Musgrave send someone else to rip the mask off Translator Lady and interrogate him about Lindsey?
Musgrave likes Ethan and probably feels that he's owed the truth. He and Ethan also have history, and even if his trust in him is resoundingly broken, he's more likely to spill to Musgrave than to a random Mook or Davian.
