Is it permissible to use housemates' belongings without prior permission, knowing in advance that they approve? And do sins such as these have negative repercussions on exam results?

Customary permission takes the place of actual permission, so there is no harm in using a friend's belongings if it is customary for him not to object to it. Furthermore, sins may be a cause for lack of success in exams and work, because success is a blessing from Allah, and a disobedient person does not deserve it due to his shortcomings, in fulfillment of His (Allah's) saying: "Whatever good befalls you is from Allah, and whatever evil befalls you is from yourself."