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.

What is the ruling on praying behind an undesirable Imam?

It is permissible to pray behind an Imam who is undesirable for committing religious violations, and it is disliked for such an Imam to lead the prayer. Still, If he was undesirable for a personal, or a worldly matter, then such argument doesn’t hold water. And Allah Knows Best.

I wish to Pray at the Mosque, but I couldn't because I am a Woman. How can I get such a Reward?

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                                                                                                                                                                                                                            Abdullah Ibn Masood (May Allah be Pleased with him) reported that the Prophet (PBUH) said: "Her prayer inside her home is better than its courtyard, and her prayer in her own bedroom is better than inside her home." [Abu Dawood]. Her staying at home, and not going to the mosque in obedience to Allah`s injunctions entails a great reward similar to that bestowed on a man for going to the mosque since the Mercy and Blessings of Allah Prevails. And Allah Knows Best.

An Arab young man had an illegitimate sexual relationship with a Christian American while she was staying in one of the Arab countries. After she returned to America, she told him that she had doubts that she was impregnated by him. It is worth pointing that it is hard for him to get a visa to the USA. What is the ruling of Sharia on this?