Can someone who begins a voluntary fast break it?
It is preferable for someone who begins an act of worship not to break it.
Allah the Exalted has said {what means}: "and let not your [good] deeds come to nought!" [Muhammad/33].
However, if a person starts a voluntary fast (nafl) and needs to break it, they are going against what is preferable, but there is no sin upon them.
What is the ruling on performing the Istikhara prayer after the Witr paryer?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
The Istikhara prayer (Prayer for seeking guidance) is a Sunnah. It consists of two units (rak’ahs) performed outside of the obligatory prayers, after which the person supplicates with the traditionally narrated du’a. It is permissible to perform it before or after the Witr prayer, as the Istikhara prayer is recommended at all times except during the disliked times—the periods in which prayer is prohibited. This is because its specific reason (the Istikhara and supplication) occurs after the prayer itself, and any prayer with a subsequent reason is not permitted during the prohibited times. It should be noted that the two rak’ahs of Istikhara are not fulfilled by performing only one rak’ah, nor by a prostration of recitation (Sajdat al-Tilawah), nor by a funeral prayer (Janazah). And Allah the Exalted 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).
Do ear and nose drops break the fast?
Yes, ear and nose drops break the fast because both the nose and ears are open passages to the body cavity.