Is it permissible for a husband to ask his wife if she left the room while he was asleep, especially after an incident where a thief entered through the window?

We do not know what you are aiming at by asking your wife whether she left the room or not. If this is a suspicion that she committed an act that entails betraying you, then the default is her innocence from this heinous act. Indulging in such thoughts is a dangerous matter, a gateway to evil, and an entrance for Satan to corrupt your life; for he is keen on disrupting families and separating loved ones. Allah Almighty says: "And do not follow the footsteps of Satan. Indeed, he is to you a clear enemy." (Al-Baqarah: 168). Muslim narrated from Jabir, may Allah be pleased with him, that the Prophet, peace and blessings be upon him, said: "Indeed, Iblis places his throne on water, then he sends forth his detachments. The one among them who is lowest in rank is the greatest in causing mischief. One of them comes and says: 'I did such and such.' He says: 'You have done nothing.' Then one of them comes and says: 'I did not leave him until I separated him from his wife.' So he draws him near to him and says: 'Yes, you are the one!'" The other matter is that you mentioned that she was the one who demanded to close the window, and you were the one who wanted to leave it open. So, from where does doubt enter your heart after all of this?