What is the ruling on someone who vowed to fast for two consecutive months, but failed to do so?
He is obliged to fulfill his vow by fasting for two consecutive months, but if he fails to do so then, it remains a debt on him until he either fulfills his vow, or dies. In case of his death, his guardian is supposed to fast on his behalf, or donate food from his estate in return for fasting, but if the guardian fails to do so then, it is all in the Hands of Allah.
Are minors obliged to fast?
A male under the age of puberty isn`t required to fast, but his guardian should order him to fast if he was able to endure the hardship until being used for fasting in general. And Allah Knows Best.
Is it permissible for a woman to shake hands with her brother-in-law?
It is forbidden for the woman to shake hands with her brother-in-law because he isn`t a Mahram (unmarriageable) of hers.
I am an author, and I have books stored with me. Is it permissible to pay my zakat in the form of books containing beneficial knowledge for poor students, such as chemistry and mathematics, or religious books that spread Islamic sciences?
Zakat on wealth must be paid in cash from the wealth itself. Therefore, you should pay your zakat in monetary form so that the poor can purchase what they need. And Allah Knows Best.