Is it permissible for a woman who has become pure before the end of forty days of confinement to perform different acts of worship? and is it lawful for her husband to have sex with her?
Once postpartum bleeding ceases and she is certain that it won`t happen again, she becomes pure, and so she should perform Ghusl (ritual bath), pray, and fast. She can also have sex with her husband even before the end of (40) days because the minimum of postpartum bleeding is a moment, and the maximum is (60) days. However, the postpartum bleeding for the majority of women is (40) days, but this does not apply to rest of them.
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.
Should the Zakah (obligatory charity) giver tell the poor recipient that this is the Zakah of his money?
No, he shouldn`t. But, the Zakah giver should make an intention in his heart that it is the Zakah of his money when paying it to the poor. And Allah Knows Best.
Does wiping over one's socks during ablution permissible?
If the socks were thick, then wiping over them during ablution is permissible so long as wiping conditions are observed. And Allah Knows Best.