Is buying shares from the Arab Bank permissible?

Buying and selling shares owned by usrious companies is impermissble. And Allah Knows Best.

Is a woman`s nephew by suckling considered a Mahram ( i.e. unmarriageable) who is permitted to accompany her to Hajj?

All that is prohibited by lineage is prohibited by suckling, and a nephew by suckling is a Mahram like a nephew by lineage ;therefore, it is permissible for him to be a Mahram for his aunt in Hajj and Umrah. 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).

Is it permissible to pay the expiation of an oath to a charitable party as it feeds the poor and needy such as Tkiyet Um Ali?

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.

There is no harm on authorizing someone to handle the expiation of another to the poor and needy, since Shafiee's jurists stated that authorizing someone for the purpose of distributing Zakah, expiation and vow is permissible as stated in [Mughni Al-Muhtajj, vol.3 pp.237]. At last, the intention of the person who intends to authorize someone else suffices in this regard. And Allah Knows Best.