Can a menstruating woman listen to the Quran from the radio or television?

Yes, a menstruating woman is allowed to listen to the Quran from the radio or television. What is forbidden for her is reciting the Quran and touching the Mus-haf, but listening is permissible—in fact, it is an act of worship.However, she should not repeat along with the Qur’an or try to keep pace with it. 

What should a person, who doubts the validity of his Wudu, or over performs it, do?

One who doubts the validity of his/her Wudu while performing it should redo it until he/she is certain of having attained purity. But, if doubt was after having performed Wudu, then he/she should not pay attention to that as doubt after the end of an act of worship does not count. This is of course in case doubt was within the reasonable limits since once it goes beyond that, it becomes a whispering of the Shaytaan (devil) which he/she should ignore as the Wudu is valid.

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).

What is the ruling of Islam on swearing by Allah without intending to take an actual oath?