Is it permissible for women to attend Tarawih prayer in the mosque?
Yes, it is permissible for women to attend Tarawih prayer in the mosque, provided they observe modesty, proper covering, and avoid mixing with men.
The Prophet ﷺ said: "Do not prevent the female servants of Allah from the mosques of Allah." [Bukhari and Muslim]
What is the ruling on congratulating others on the arrival of Ramadan?
Congratulating others on the arrival of Ramadan is permissible.
If someone does it to express their love for worship and the blessings of Ramadan, and congratulates their fellow Muslim, they have done something good.
The Prophet ﷺ said: "Shall I not tell you something that, if you do it, you will love one another? Spread peace (Salam) among yourselves." [Narrated by Muslim]
Giving Salam is a prayer for peace and safety.
What is the ruling on performing the obligatory prayer immediately after the commencement of the Adhan?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is permissible to perform the obligatory prayer (Fard) immediately after the commencement of the prayer time. As the Adhan (call to prayer) serves as a sign that the time has entered, it is permissible to pray as soon as the Adhan begins. However, it is recommended (Mustahabb) for whoever hears the Mu’adhin to repeat what he says. If the Adhan occurs while one is already engaged in prayer, he should repeat the words of the Mu’adhin after he has completed his prayer. And Allah the Exalted knows best.
What is the ruling on wiping the front of the head beneath the ḥijāb, and is it permissible to wipe over the ḥijāb if it was put on while in a state of purification (wudu`)?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
It is permissible to wipe the front portion of the head (nāṣiyah) with the fingertips. It is likewise sufficient to place a wet hand over a head covering (ḥijāb), provided the moisture actually reaches the hair beneath it — if it does not, it does not suffice. A head covering is not treated in the same manner as leather socks (khuff) and may not be wiped over in lieu of the head itself.
Imām al-Nawawī, may Allah have mercy upon him, states in al-Majmūʿ (Vol.1/P.407): "If a person is wearing a turban and does not wish to remove it — whether for a valid reason or otherwise — he should wipe the entire front portion of the head... The same ruling applies to whatever a woman wears on her head. If, however, he confines himself to wiping over the turban without wiping any part of the head itself, this does not suffice — and there is no disagreement among us on this point." He further states: "A woman is like a man in the manner of wiping the head... She should insert her hand beneath her head covering so that the wiping falls upon the hair itself. If she places her wet hand over her head covering, our scholars stated: if the moisture does not reach the hair, it does not suffice her." And Allah the Almighty knows best.