Is it permissible for the person who hasn`t made wudu` (state of minor ritual impurity) to perform the Adhan?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is disliked (Makruh) for a person in a state of minor ritual impurity (Hadath Asghar) to perform the Adhan. However, if he does so, his Adhan is considered valid and fulfills the sunnah of the Adhan despite it being disliked. And Allah the Exalted knows best.
What is the meaning of the Prophetic statement that a boy is held in pledge (murtahan) for his 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
The first interpretation: That if the boy dies in infancy without an 'aqīqah having been performed on his behalf, he will not intercede for his parents on the Day of Resurrection. This is the position of Imam Aḥmad ibn Ḥanbal, and Imam al-Khaṭṭābī concurred with him, stating: "The finest of what has been said regarding this matter is the position adopted by Aḥmad ibn Ḥanbal." — [Fatḥ al-Bārī by Ibn Ḥajar,{Vol.9/P.594]
The second interpretation: That the child is likened to a pledged object (marhūn) — one from which full benefit and enjoyment cannot be derived until it is redeemed. A blessing is only made complete upon the one blessed when they fulfil the obligation of gratitude (shukr), and the prescribed expression of gratitude for this particular blessing is what the Prophet ﷺ established as Sunnah — namely, the slaughtering of the 'aqīqah on behalf of the newborn as an act of thankfulness to Allah the Almighty and as a supplication for the wellbeing and safety of the child. This is the position of Mullā 'Alī al-Qārī. See: [Mirqāt al-Mafātīḥ Sharḥ Mishkāt al-Maṣābīḥ, Vol.7/P.2688]
And Allah Almighty knows best.
Is it permissible for a wife to give her money to her family as a charity, or a gift without asking her husband, or seeking his consent?
The wife has the right to give her money as a charity, or a gift to her family, or to other people after consulting her husband out of respect, and this is the meaning of treating on footing of kindness and equity. Therefore, if he wanted to stop her from helping her family, then there is no harm in not telling him.
What is the expiation for perjury?
Perjury is forbidden and one of the major sins that require turning to Allah in repentance, seeking His forgiveness, giving back rights to whom they belong, or seeking their forgiveness, and expiating for that oath.