Is it permissible to swear by (Allah "The Living")?
Yes, it is permissible because "The Living" is one of Allah`s Attributes.
What is the ruling on staring at the Holy Ka'abah?
All perfecy 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.
Staring/looking at the Holy Ka'abah is an act of worship even if this took place while performing a prayer, since Al-Baihaqee stated that the Prophet (PBUH) said: " Staring at the Holy Ka'abah is an act of devotion." [Sho'aab Al-Imaan]. And Allah Knows Best.
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).
I have 490 shares in a commercial company, and they have been held for a year. The price of each share is one dinar, noting that the shares are at a loss. I want to know the amount of zakat due on them?
Commercial shares are subject to zakat based on their current market value, even if they are at a loss. The zakat rate is 2.5%. And Allah Knows Best.