What is the duration one must wait after the Dhuhr Adhan – and the Adhan for other prayers – to perform the prayer, and is it true that there is a half-hour period after the Adhan during which prayer is not permissible?

There is no specific waiting period after the Athan (call to prayer) for Dhuhr prayer; rather, it is permissible to pray as soon as the time for the Athan begins. Performing the prayer at the beginning of its time is generally better, based on the Hadith: "Prayer at its appointed time." It is recommended to delay the Dhuhr prayer when the heat is intense. There is no prohibition against praying immediately after the Athan. The prohibition is only against praying at the zenith (midday) before the time for Dhuhr begins, and this prohibition does not include obligatory prayers.