Is it obligatory to lower one's voice or hasten the prayer when family members enter the room due to the lack of privacy, and is there any harm in that?

It does no harm if your family or others hear your recitation during the night prayer (Salat al-Layl). The Sunnah is to recite aloud in it, and the matter is broad, allowing for both loud and silent recitation. However, it is emphasized to lower the voice if raising it would disturb others. The night prayer holds immense virtue; it is the practice of the righteous, a means of drawing closer to Allah, an expiation for sins, and a deterrent against illness.