Is the marriage contract considered valid if concluded at home by the marriage official (Ma`zon)?
Yes, it is valid as long as it is registered at the court.
Is it permissible for a woman to adorn herself with kohl (eyeliner) and go out wearing it?
Kohl is a form of adornment, and it is forbidden for a woman to display her adornment to anyone other than her husband or a Mahram. And Allah Knows Best.
Is it permissible to divide the Aqeeqah amongst one`s brothers and family?
Giving a portion of the Aqeeqah, even a small one, to the poor and needy is obligatory, and if one`s brothers and family are among the needy then, they are more entitled to it, and this way it is a charity and an observation of kinship ties. However, if they aren`t needy then, it is permissible to give them from the Aqeeqah after giving the poor and needy their share.
How should a woman make up prayers that she missed two days from delivery?
All perfect praise be to Allah,The Lord of The Worlds It is prohibited to delay prayer beyond its prescribed fixed time, therefore, the woman in question is obliged to make up missed prayers after delivery and confinement i.e. once she becomes ritually pure. And Allah Knows Best.