He had an affair with his mother's sister's daughter and they both had an intercourse and that lead to a 6th months pregnancy and later both his mother and aunty got settle the matter with mutual understanding and arranged the Nikaah of both girl and boy, while Nikaah girl was conceiving 6th month pregnancy and later they decide not to do any Vidayee (Valima), after that the did not continue there relation and neither there elders allowed to do so. And the boy he waited for the answer from the girl's family and girl she delivered the baby, and after that he finally took a decision and gave Talaaq to girl.
I need the proper answer to the following questions and the above story.
1) Is it a legal marriage according to Islaamic law?
2) Is the marriage completed without a Vidayee or Valima?
3) Is it a Talaaq applicable on this Nikaah?
4) If this is not a Nikaah. Can they again perform Nikaah?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

Too much tolerance between a man and a marriageable woman so that the religious requirements are not met, like the woman not wearing Hijab and being in seclusion with a man, may lead to many evils, such as you mentioned. That is why Islam forbade such relationships, so those who do this must repent to Allaah from this sin. As regards the questions that you asked, the answers are as follows:

First point: It is not permissible for an adulterer or fornicator to marry the woman he committed adultery or fornication with except with the following two conditions:

(1)   To repent

(2)   To make sure that the woman is not pregnant, and if she is pregnant, he must wait until she gives birth to the child.

This is the most preponderant opinion. Therefore, you know that the marriage that took place is invalid.

Second point: A woman becomes a wife as long as the marriage contract is concluded, on the condition that the marriage contract is correct, even if the marriage is not consummated or the marriage banquest is not organized.

Third point: The divorce that takes place after a marriage that is Islamicly controversial, is considered as one divorce (of the three), according to some scholars.

Fourth point: It is permissible for this man to marry his maternal cousin with whom he committed fornication if the two above conditions (in first point) are met, even if he had already divorced her.

Allaah knows best.