From the time of azan of zuhr upto sunset is time for both zuhr and asr. If you would like to pray at fazilat time when the reward is greater, for zuhr, it is at azan of zuhr and for asr, it is at azan of asr. As for the time for maghrib and Isha, it is from maghrib till midnight (which is calculated as per Islamic law). Fazilat for maghrib namaz is when the from when the azan is recited and around an hour after maghrib, when Isha azan is recited, it is time for Isha prayers when the reward is greater.
You pray once the Azan for Zhur is done. You don’t have to wait for Asr. Likewise in the case of Magrib