Is it permissible to offer mandatory prayer while sitting?
All perfect praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions. Standing up is a pillar in the mandatory prayer and the prayer is not valid without it unless one cannot do it. As for Nafila (supererogatory prayer), one can perform it in the state of sitting even if he/she can stand, but his/her reward is half the reward of the person who prays while standing if he/sh has no legal excuse for such an act. And Allah Knows Best.
Is it permissible for a wife to give her money to her family as a charity, or a gift without asking her husband, or seeking his consent?
The wife has the right to give her money as a charity, or a gift to her family, or to other people after consulting her husband out of respect, and this is the meaning of treating on footing of kindness and equity. Therefore, if he wanted to stop her from helping her family, then there is no harm in not telling him.
Is it permissible for a woman to ride a taxi without a Mahram (unmarriageable kin) in order to attend a gathering of knowledge?
It is permissible for the woman to pursue the knowledge that she needs in order to perform the religious requirements due on her if there was nobody to teach her at home, provided that there is no temptation involved. Moreover, we don`t recommend that she rides a taxi without a Mahram although such an act isn`t from the forbidden Khulwah (seclusion).
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.
