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 none provided for particularly by whom are obliged to provide for her. And Allah Knows Best.
What is the ruling on the ablution of one who cuts his nails, and is it permissible to cut them before the ritual bath for major impurity (janabah)?
Cutting nails does not invalidate ablution, and it is permissible to cut them before the ritual bath for major impurity. And Allah the Almighty knows best.
What is the ruling on using eye drops while fasting?
Eye drops do not break the fast, even if the taste reaches the throat, because the eye is not an open passage to the body cavity.
Is a woman sinful if she fasts out of shyness from her family while menstruating or in postnatal bleeding?
It is forbidden for a woman who is menstruating or in postnatal bleeding to abstain from food and drink with the intention of fasting. If she fasts out of shyness, she is sinful, as her fast is not valid.
However, if she refrains from eating and drinking without intending to fast, it is not forbidden, but she unnecessarily burdens herself without need.