Is it permissible for a woman to offer the obligatory prayer after Adhaan (call for prayer), and before Iqamah (The announcement that the prayer is about to begin) in the mosque?
Yes, it is permissible for a woman to offer the obligatory prayer immediately after the Adhan and before the Iqamah in the mosque. And Allah Knows Best.
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 are the key differences between the 'aqīqah and the uḍḥiyyah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
The following are the key differences between the 'aqīqah and the uḍḥiyyah:
First: The 'aqīqah is slaughtered as an act of drawing closer to Allah the Almighty and expressing gratitude for the blessing of a newborn child. The uḍḥiyyah, on the other hand, is slaughtered as an act of drawing closer to Allah and expressing gratitude to Him specifically during the days of slaughter (ayyām al-naḥr).
Second: The 'aqīqah is performed on the seventh day from the birth of the newborn, whereas the uḍḥiyyah is performed on Eid al-Aḍḥā and its time extends for three days after the Eid.
Third: The 'aqīqah is performed once in a lifetime for the newborn, whereas the uḍḥiyyah is recommended every year.
Fourth: It is Sunnah for the one intending to offer the uḍḥiyyah to refrain from cutting his hair and nails until after he has slaughtered. This is not a Sunnah for the one intending to perform the 'aqīqah.
Fifth: It is Sunnah for the 'aqīqah to be cooked and distributed to the poor in its cooked form. The uḍḥiyyah, by contrast, must be distributed as raw meat. And Allah Almighty knows best.
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.