Is it permissible for a woman to sit with her brothers-in-law?
It is impermissible for a woman to sit with her brothers-in-law, and it is also impermissible for a person to have a Khulwa (seclusion) with his sister-in-law.
Should a woman who broke her fast because of delivery make up for missed fasting days before the next Ramadhaan, and what is the expiation due on her in case she delayed making up for them ?
She should make up for missed fasting days before the start of next Ramadan if possible, but if she didn`t while being able to, then she is obliged to make up for them along with feeding a needy person for each delayed day of the missed fasting days. However, if she wasn`t able to make up for the missed fasting days before the start of next Ramadan, she has to fast a day for every day that she missed, and no ransom is due on her. And Allah Knows Best.
Is it permissible to slaughter one animal as a sacrifice and an Aqeeqah (the sheep slaughtered on the seventh day from the child`s birth)?
It is impermissible to do so since each of them is slaughtered for a different reason.
What should one who made an intention, at night, to make up for a missed fasting day, but broke his fast on the next day?
One who started making up a missed fasting day, then broke his/her fast without a legitimate reason is considered sinful, and is only obliged to make up for the missed fasting days. And Allah Knows Best.