Is it permissible for a woman who broke fasting due to delivery to pay a ransom ?
She should make up for the missed fasting days once she becomes ritually pure, but it is impermissible for her to pay a ransom while being able to fast.
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 Ramadhaan 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 Ramadhaan, she has to fast a day for every day that she missed, and no ransom is due on her.
Is wearing a cap permissible for boys, or girls?
All Perfect Praise is due to Allah, The Lord of The Worlds.
In principle, anything is permissible sol long as there is no Sharia evidence prohibiting it, thus wearing the cap by boys is permissible as long as it isn`t done in imitation of the disbelievers. Moreover, a Muslim woman should avoid all that which attracts the attention of men because whatever leads to the forbidden is forbidden. And Allah Knows Best.
Is it permissible for a person to give the Zakah(obligatory charity) to his daughter-in-law?
Yes, it is permissible for a person to give the Zakah to his daughter-in-law if she was poor, and wasn`t provided for sufficiently by whomever is obliged to provide for her.