What If You Pray Before Adhan by Mistake
Question

1 – We were travelling and we stopped in a city on the way, where we prayed Dhuhr and `Asr when the time for Dhuhr began, and when we had finished praying, the Mu’dhdhin gave the Adhan for Dhuhr prayer. Do we have to repeat the prayer or not? 
2 – We were heading towards a city in the north, and after we had stayed there overnight, we formed the intention to leave and we prayed with the intention of shortening and joining our prayers. We prayed Dhuhr, then when we said Takbir for ‘Asr, the Mu’dhdhin gave the Adhan for Dhuhr but we completed our ‘Asr prayer. Do we have to repeat the Dhuhr and ‘Asr prayers or do we have to repeat Dhuhr only? Please note that we are now at home and are no longer travelling. 
3 –I entered the mosque to pray Dhuhr and the prayer had finished, so we prayed in congregation but no one said Takbir for a prayer in congregation. What should I do?
Praise be to Allah.Can we pray before Adhan?
The answer to the first and second questions is that if the Mu’dhdhin gave the Adhan for Dhuhr on time, and was not late in doing so, then you offered the prayer before it was due , so you have to repeat Dhuhr and `Asr. The same applies to anyone who prays then finds out that the time for the prayer had not yet begun. 
Ibn Qudamah (may Allah 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 Adh-Dhuhri, Al-Awza`i, Ash-Shafi`i and Ashab Ar-Ra’i (i.e., Hanafies). It was also narrated that Ibn `Umar and Abu Muosa repeated Fajr because they offered it before the time for it began.”  (Al-Mughni, 2/45) 
The basic principle is that the Mu’dhdhin does not give the Adhan 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 Mu’dhdhin 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 Mu’dhdhin 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 Qudamah  (may Allah have mercy on him) said: 
“If a person hears the Adhan from one who is trustworthy and knows the time, he may rely on him, because it seems that he does not say the Adhan until after the time for prayer has begun, so it is as if his Adhan announces the beginning of the time. The Prophet (peace and blessings of Allah be upon him) said: “The Mu’dhdhin is a position of trust.” (Narrated by Abu Dawud) Were it not that he is to be relied upon, he would not be in a position of trust. Moreover, because the Adhan is prescribed so that the time may be known, if it were not permissible to rely on the Mu’dhdhin then there would be no point in prescribing the Adhan. The people still gather in their mosques at the times of prayer, and when they hear the Adhan they get up to pray, based on the Adhan of the Mu’dhdhin without checking the time, so there is consensus on this matter.” (Al-Mughni, 2/31) (See also Al-Majmu`, 3/79) 
Can you pray without Takbirat Al-Ihram or the Iqamah?
With regard to your question about praying without saying Takbir, if what you mean is that the Imam did not say Takbirat Al-Ihram (Takbir 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 Iqamah for the prayer, or that the Imam did not say the Takbirat Al-Ihram out loud, then the prayer is valid, praise be to Allah, because the Iqamah 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 Takbirat Al-Ihram out loud. 
And Allah knows best.