Do unintentional naps for a few minutes—due to extreme drowsiness while in the restroom to perform the Fajr prayer—cause possession by jinn or devils, even if one recites the supplication for entering the restroom most of the time?

We do not know of any evidence from the Sunnah that links forgetting to mention Allah's name (dhikr) with a jinn entering a human. However, one must say the dhikr before entering the lavatory, rely upon Allah, and not prolong one's stay in the lavatory.