Is it Permissible to lighten the eyebrows (I.e. Dye them blond)?
All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.
Lightening the eyebrows is applying dye to them, and it is permissible to dye hair with other than black. The proof of this is the Hadith related by Muslim on the authority of Jabir told that when Abu Quhafa (Abu Bakr’s father) was brought on the day of the Conquest of Mecca with his head and beard white like hyssop, the Prophet (PBUH) said: "Change this with something, but avoid black."
However, it isn`t permissible for a Muslim woman to show her ornament (Zeenah) in the presence of non-Mahram men (Men permissible for a woman to marry). And Allah The Almighty Knows Best.
Is it permissible for the mother to offer an Aqeeqah (the sheep slaughtered on the seventh day from the child`s birth)for her son?
Aqeeqah is due on the one obliged to provide for the newborn, and it is permissible for the grandfather, or the mother to offer the Aqeeqah.
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.
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.
