Is naming specific days and nights with special names based on anticipated weather conditions (such as "Qurrat al-Anz" for days of extreme cold) considered a claim to knowledge of the unseen, or is it based on experience and knowledge of cosmic laws? What is the ruling on naming them and writing about them in research nowadays?

These designations have no relation to claiming knowledge of the unseen. People's awareness of the occurrence of cold in a specific season is not a claim to knowledge of the unseen; rather, it is an awareness of what has become the norm according to Allah's established ways. People discern the seasons of winter and summer based on recurring natural phenomena. Similarly, the naming of Arabic months was based on events that occurred within them, such as heat, like the naming of Ramadan from the intensity of the "Ramdaa" (scorching heat). Therefore, there is no harm in naming these seasons according to what has been conventionally agreed upon, because it falls under the category of naming a period of time based on what Allah's established way has caused to occur within it.