The answer to the first and second questions is that if the muezzin gave the adhaan for Zuhr on time, and was not late in doing so, then you offered the prayer before it was due, so you have to repeat Zuhr and ‘Asr. The same applies to anyone who prays then finds out that the time for the prayer had not yet begun.
Ibn Qudaamah (may Allaah have mercy on him) said: If a person prays before the time (for that prayer) begins, it does not count, according to the view of the majority of scholars, whether he did that deliberately or by mistake, all of the prayer or part of it. This was the view of al-Zuhri, al-Awzaa’i, al-Shaafa’i and ashaab al-ra’i. It was also narrated that Ibn ‘Umar and Abu Moosa repeated Fajr because they offered it before the time for it began. End quote from al-Mughni, 2/45.
The basic principle is that the muezzin does not give the adhaan until after the time for the prayer has begun, especially in a case such as yours, when you have no evidence to suggest that the muezzin made a mistake about the time or was not careful about it. Rather the fact that you did this again indicates that you were not very careful about checking the beginning of the time, so you should have relied on the muezzin in order to know when the time began, especially since the people of the place where you were relied on him, and there is nothing to suggest that they objected to him.
Ibn Qudaamah (may Allaah have mercy on him) said: If a person hears the adhaan from one who is trustworthy and knows the time, he may rely on him, because it seems that he does not say the adhaan until after the time for prayer has begun, so it is as if his adhaan announces the beginning of the time. The Prophet (peace and blessings of Allaah be upon him) said: “The muezzin is a position of trust.” Narrated by Abu Dawood. Were it not that he is to be relied upon, he would not be in a position of trust. Moreover, because the adhaan is prescribed so that the time may be known, if it were not permissible to rely on the muezzin then there would be no point in prescribing the adhaan. The people still gather in their mosques and jaami’s at the times of prayer, and when they hear the adhaan they get up to pray, based on the adhaan of the muezzin without checking the time, so there is consensus on this matter. End quote from al-Mughni (2/31). See also al-Majmoo (3/79).
The answer to the third question is: with regard to your question about praying without saying takbeer, if what you mean is that the imam did not say the takbeerat al-ihraam (takbeer to start the prayer), then his prayer and the prayer of those who prayed behind him is not valid, and you all have to repeat the prayer.
But if what you mean is that you did not say the iqaamah for the prayer, or that the imam did not say the takbeerat al-ihraam out loud, then the prayer is valid, praise be to Allaah, because the iqaamah is not one of the essential parts of the prayer or one of the conditions of prayer being valid. The same applies to the imam saying the takbeerat al-ihraam out loud.
And Allaah knows best.