Is moving the lips without uttering a sound considered an utterance by which a vow (nadhr) is contracted? And is it obligatory to fulfill an absolute vow immediately? And are studies and exams considered an excuse to delay fulfilling a vow if it is a fast?

A vow (nadhr) is not binding unless it is uttered verbally, and the minimum for this is for one to hear oneself, provided there are no impediments. Shaykh al-Islam is of the view that merely articulating the letters from their proper articulation points is sufficient. If you doubt its validity, the default is the absolution of your responsibility. We advise you to abandon vowing and not to succumb to obsessive thoughts.

As for fulfilling an unrestricted vow (nadhr mutlaq), there is a difference of opinion among scholars: The Hanafis, Malikis, and some Hanbalis are of the view that its fulfillment can be delayed, because it is an open-ended obligation. The Shafi'is and some Hanbalis are of the view that it must be fulfilled immediately. This latter opinion is more cautious and absolves one's responsibility more effectively.