What are the nights on which intercourse is not permissible in the Islamic months, and is it true that intercourse is not permissible upon sighting the crescent moon at the beginning of the month, according to the hadith?

There is no validity to the claim that intercourse is impermissible upon sighting the new moon at the beginning of the month. It is permissible at all times, except in specific situations: while in a state of ihram (definition) for Hajj or Umrah (definition), while fasting during the day, and during a woman's menstruation or post-natal bleeding (nifas (definition)).