Is it permissible to break the fast immediately upon hearing the Adhan (call to Salah) of Maghrib (Sunset) Prayer during Ramadan, or must one wait until the Mu'adhin (caller to Prayer) finishes the Adhan and then one may break the fast?
One should break the fast when seeing the sunset, or by hearing the Adhan of the Mu'adhin who raises the Adhan only after the sun has set. Thus, one should break fast upon hearing the Adhan.
Is it permissible to drink while the Mu'adhin is still in the middle of raising the Adhan for Fajr (Morning) Payer during Ramadan?
It is incumbent upon the fasting person to intend the fast when clearly seeing that the dawn time has fallen or when (Part No. 9; Page No. 28) hearing the Adhan of the Mu'adhin who thoroughly makes sure that the Fajr time has fallen before making the Adhan. Allah (Exalted be He) said:
...and eat and drink until the white thread (light) of dawn appears to you distinct from the black thread (darkness of night)
May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family, and Companions.