1. Is it obligatory to repeat the Dhuhr and Asr prayers if they were performed in a city during travel at Dhuhr time, then the muezzin called the Adhan for Dhuhr after the completion of the prayer? 2. Is it obligatory to repeat the Dhuhr and Asr prayers, or only Dhuhr, if they were prayed shortened and combined before leaving a city, then the muezzin called the Adhan for Dhuhr during the Takbir of Asr? 3. What is incumbent upon someone who entered the mosque for Dhuhr prayer and found that he had missed the congregational prayer, so he prayed in a congregation for which no one made the Takbir?

If the muezzin called the adhan (definition) for Dhuhr at its proper time, and you prayed before the time entered, then you must repeat Dhuhr and Asr, and every prayer you prayed before its time entered, for Ibn Qudamah stated that whoever prays before the time, their prayer is not valid.

The basic principle is that the muezzin does not call the adhan until after the time has entered. It is obligatory for you to rely on the muezzin to know when the time has entered, especially since the people of the place rely on him and have not objected to him. The muezzin is entrusted, and it is permissible to follow him.

As for your prayer without saying the takbir, if the imam did not pronounce the takbirat al-ihram (definition), then neither his prayer nor the prayer of those behind him is valid, and you must repeat it. If what is meant is that you did not pronounce the iqamah (definition) for the prayer or that the imam did not vocalize the takbirat al-ihram, then the prayer is valid, because the iqamah and vocalizing the takbirat al-ihram are not among the pillars of prayer or conditions for its validity.