Praise be to Allah.
The Sunnah with regard to Taraweeh prayer is to say the tasleem after every two rak‘ahs.
With regard to praying it four rak‘ahs by four, this is contrary to what is preferable, although a number of the scholars regarded it as permissible, and others regarded it as mustahabb, as is the view of the Hanafis. We have previously discussed the differences of scholarly opinion concerning this matter in the answer to question no. 195355.
If the imam does something that is valid according to some of the scholars, or based on his own ijtihaad, or following the opinion of a scholar, or because he thinks that it is correct, then his prayer is valid and the one who is praying behind him must follow him.
Praying (Taraweeh) in the mosque with the imam is better, and if you pray with the imam until he finishes, the reward of spending an entire night in prayer will be recorded for you.
With regard to the difference in the time of the adhaan for Fajr – in Turkey – between how it was before Ramadan and how it is in Ramadan, the reason for that is that our Turkish brothers delay the adhaan throughout the year so that they can pray Fajr when the light spreads more in the horizon (after the break of true dawn but before sunrise), as is recommended according to the Hanafi madhhab, but they give the adhaan for the correct time of Fajr in Ramadaan so that people can stop eating when dawn breaks.
Hence you have to stop eating when you hear the adhaan, and you have to pray Fajr with the congregation in the mosque.
For more information, please see the answer to question no. 66202
And Allah knows best.