Is it permissible to be absent from a funeral prayer due to exams and the distant location of the burial, and how can I avoid the punishment for that, and is the prayer of al-Wahsha (solitude) that is offered for the deceased on the night of burial valid?

First: Friendship between a man and a non-mahram woman is not permissible due to the evil and temptation it entails.

Second: The funeral prayer (Salat al-Janazah) is a communal obligation (fard kifayah); if some perform it, the obligation is lifted from the rest. Missing it due to an excuse, such as difficulty in reaching the location or examinations, does not result in any consequence. The statement that a house from which no one attends a Muslim's funeral will have everyone in it punished has no basis. The intent of 'fard kifayah' here is if all Muslims abandon praying over the deceased, not singling out a specific house.

Third: "Salat al-Wahshah" (Prayer of Loneliness/Solitude) is an innovated prayer unknown to the أهل السنة (People of the Sunnah) and not found in their books. Rather, it is known among the Imami Shiites, who narrate a fabricated hadith concerning it. Suffice it as proof of its invalidity that it has no origin among the Ahl al-Sunnah, but is found in the books of the people of innovation.