Is it obligatory to fulfill the aforementioned vow concerning looking at forbidden things, looking at X-ray images on the inspection machine at the airport, and writing a bad word on a website, given circumstances such as studying and not remembering the details of some incidents?

The vows mentioned are vows of 'lajaj' (obstinacy or contention), and what is obligatory concerning them is one of two options: fulfilling the vow or performing an expiation for an oath (kaffarat yamin (definition)). If you have broken any of these vows, then the expiation becomes binding upon you. It is important to note that casting one's gaze upon what Allah has forbidden is prohibited, and looking at the entire body of a non-mahram (definition) woman is forbidden, except for what the Sharīʿah has exempted. One must be wary of following the footsteps of Satan, and it is obligatory to abandon disobedience to Allah and to distance oneself from what displeases Him. The day begins with the rising of dawn and ends with the setting of the sun.