Is it permissible for a woman to perform Fajr prayer immediately upon hearing the adhan, and is it permissible to continue eating and drinking during the Fajr adhan while fasting, based on the premise that the current Fajr adhan is based on investigation and ijtihad and may be before the actual time?

The decisive factor for Fajr prayer, abstention (imsak), and breaking the fast (iftar) is the rising of dawn and the setting of the sun. If a person is certain that the calendars contradict this, then they are not to be relied upon. However, if the matter is merely based on rumors, then they should be disregarded, as this creates confusion among Muslims and disrupts their acts of worship.