Is it obligatory to prioritize attending a funeral over being preoccupied with studies and searching for medical treatment for relatives? And does greeting those present with "peace be upon you" before the start of the funeral procession contradict the required silence?

There is no harm in missing a funeral procession due to being busy, because it is a communal obligation (fard kifaya). Witnessing the funeral, praying over it, and burying it carries great reward, as the Prophet, peace and blessings be upon him, said: "Whoever attends a funeral until he prays over it, will have a qirat; and whoever attends until it is buried, will have two qirats." The two qirats are like two great mountains.

As for spreading peace, there is no harm in it. However, raising one's voice with dhikr (remembrance of Allah) and tahleel (uttering "La ilaha illallah") behind the funeral for the purpose of worship is disliked, as this is an innovation (bid'ah). The Sunnah for one who follows a funeral is silence. He should not raise his voice with recitation, dhikr, nasheed, or anything else. The companions of the Prophet, peace and blessings be upon him, disliked raising voices near funerals. Moreover, it resembles the Christians. The wisdom behind it is to calm one's thoughts and to gather one's focus regarding the funeral.