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.
What is the ruling on public street water if it gets on a person's clothes or body?
The default ruling is that public street water is pure (tahir). If one is certain of its impurity (najasa), then a small amount that gets on the lower part of a person's clothes or body is overlooked (excused). And Allah the Almighty knows best.
How is the meat of the 'aqīqah to be distributed?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
If the 'aqīqah is a recommended one (mandūbah), it is obligatory to give some portion of it in charity to the poor — even if the amount is small — with the minimum being approximately half a kilogram of meat.
The most preferable manner of distribution is to divide it into three equal portions, as with the uḍḥiyyah:
One third for the guardian and his household to eat from
One third to be given in charity to the poor
One third to be gifted to friends and neighbours, even if they are wealthy
It is furthermore more preferable to send the food to the poor already cooked, rather than inviting them to come and eat. And Allah Almighty knows best.
If a father stops his daughter from getting married, is it permissible for her to conclude the marriage contract herself?
It is permissible for the father to stop his daughter from getting married if there was a lawful reason for that, and she isn`t allowed to conclude the marriage contract without her guardian. However, if her father denied her right in getting married for an unlawful reason, then she should go to court.