(A) Do parents have any right over the marriage of their children?

The father and paternal grandfather have full authority over the children who have not yet reached the age of puberty.

When a child reaches the age of puberty, then there are three different situations:

(a) a child who is baligh but not mentally mature: in this case, the father and the grandfather still have their authority over him or her. Such a child cannot take a decision on marriage without the approval of the father or the grandfather.
(b) a male child who is baligh and also mentally mature: in this case, he has full right to decide about his own marriage.
(c) a girl who is baligh and mentally mature: There are four different opinions on this issue. (Muhaqqiq al-Hilli, Sharaya', p. 500: Shahid ath-Thani, Sharh Lum'ah, vol. 2. p. 71. ) But the majority of the present mujtahids say that in her first marriage, a baligha and mentally mature girl cannot marry without the permission of her father or grandfather. And if such a girl had already married before. then she has full right to decide for herself in the case of her second marriage. (Al-Yazdi. al-'Urwah, p. 654; al-Khui, Minhaj, vol. 2. p 255: al-Khumayni. Tahrir, vol. 2. p. 254; for details, see ash-Shahid ath-Thani, Masalik, vol. 1, p 449-452. )

This law is in place to act as an extra protection for the honor of a teenage girl. However, if the father or grandfather refuses to give his consent to a proposal of a suitable man, then a religious judge can over-ride the decision of her father or grandfather. Similarly, if the father or grandfather is inaccessible, then there is no need for his permission.

(B) By considering the hijab in Islam, is a boy permitted to look at the girl before marrying her?

A man is allowed to see without hijab the woman whom he intends to marry. However, this permission is just for the man or the woman to see the prospective spouse once; (Muhaqqiq al-Hilli, Sharaya', p. 493; Shahid ath-Thani, S'harh Lum'ah, vol. 2, p. 67; al-Yazdi, al-'Urwah, p. 625; al-Khu'i, Minhaj, vol. 2, p. 253; al-Khumayni, Tahrir, vol. 2, p. 254: 'Allamah al-Hilli, Tazkirah, vol. 2, p. 572; Shaykh at-Tusi, an-Nihayah, p. 484.) it is not a perpetual license to go out together! I strongly recommend that such a meeting should take place with the supervision of the parent or the guardian.
(C) Can a boy and a girl who are engaged meet each other or go out together ?

In Islam, there is no ceremony known as engagement. According to the shari'ah, engagement is a revocable agreement between two persons to marry each other, nothing more; it does not make the two persons mahram to each other. They still have to observe the rules of hijab. But if two persons who are engaged wish to meet each other, then the only way to legalize that Islamically is by performing mut'a marriage between the fiance and the fiancee. They can even put a condition in the mut'a that there will be no sexual relations. This way, they will be able to meet each other without any objection from the shari'ah. This method can be adopted by those also who wish to go through the engagement ceremony in which the fiance puts a ring on the fiancee's finger.

However, as mentioned earlier, a girl who has not married before will need her father's or grandfather's approval even in such a mut'a marriage.