Is there any religious prohibition against setting aside Friday for reading Prophetic hadiths, or concluding a gathering by reciting verses from the Holy Quran without specifying particular verses?

There is no harm in dedicating Friday for a Hadith study session if it is because it is a day off from work, and not out of a belief in the superiority of holding the session on that day. This is because dedicating an act of worship to a specific time that the Sharia has not singled out, while believing in its superiority, is an innovation (bid'ah). Such dedication is permissible if it is based on a reasonable cause, such as having free time, but it is not permissible if it becomes a pretext for believing in something that is not prescribed. As for concluding a Hadith study session with the recitation of verses from the Holy Quran, it is permissible due to the general virtue of reciting the Quran and gathering for it.