Which is more excellent for a woman: to pray in the mosque or in her house?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
Our Master the Prophet ﷺ said: 'It is more excellent for a woman to pray in her house than in her courtyard, and more excellent for her to pray in her private chamber than in her house.' Therefore, a woman’s adherence to her home and her refraining from going to the mosques—in obedience to the command of Allah the Exalted—attains a great reward and abundant recompense. Furthermore, women praying in congregation within their homes is better than their attendance at the mosques, based on the aforementioned Hadith. And Allah the Exalted knows best.
Can I start fasting based on the moon sighting of another country?
You must follow the moon sighting of the country you are in.
What is the ruling if someone wipes his head then shaves it - must he repeat it?
If a person performing ablution shaves his head after wiping it during ablution, he is not required to re-wipe his head or repeat the ablution. And Allah the Almighty knows best.
What is the ruling on ablution without istinja' after sleep?
The sleep of one whose buttocks are not firmly seated on the ground invalidates ablution. Whoever wakes up and wants to pray must perform ablution only. Istinja' is not a condition for the validity of ablution, because it is for removing impurity from the private part. If there is no impurity on the private part, then istinja' is neither obligatory nor recommended. And Allah the Almighty knows best.