Is it permissible for a wife to leave her house without asking her husband?
It is impermissible for the wife to leave her house without asking her husband except for a sound reason.
Is it permissible to shorten the eyebrows if they get too long?
Yes, it is permissible, but not desirable.
If the Prophet (PBUH) was an Arab citizen in 2009, born in an Arab country ruled by an Arab ruler, saw what was happening to Gaza, what would he (PBUH) have done?
All perfect praise be to Allah, The Lord of The Worlds. May His peace and blessings be upon our Prophet Muhammad and upon all of his family and companions.
Had the Prophet (PBUH) been amongst us, Muslims would have supported him, and then he would do whatever pleases the heart of every Muslim. And Allah The Almighty Knows Best.
Is it permissible for a Muslim to slaughter an Aqeeqah on behalf of someone else, and offer it to him as a gift?
In principle, the guardian is the one who should offer the Aqeeqah(the sheep slaughtered on the seventh day from the child`s birth) because he is obliged to provide for the newborn, and it is impermissible for anyone else to slaughter it on his behalf unless with his consent. However, it is permissible for a person to offer the sheep, or its price as a gift to the guardian of the newborn, and then the latter can slaughter it, or deputies someone else to do that on his behalf.