What is the ruling on performing voluntary prayers during official working hours?

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 if  there was break time during official working hours, but if not then it is impermissible. 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 Aqeeqah due on the miscarried embryo?

No Aqeeqah is due on the dead embryo.

Zakah due on a woman`s jewelry?

Zakah(obligatory charity) isn`t due on a woman`s jewelry used for ornament, provided that the value of each peace is within the range common amongst people.