Is the son of the sister by breastfeeding considered a mahram in Islam, and is it permissible for a woman to go for Hajj with him?
Everything that is prohibited by lineage is also prohibited by breastfeeding. The son of the sister by breastfeeding is considered a mahram, just like the son of the sister by blood, and it is permissible for him to be her mahram for Hajj and Umrah. And Allah Knows Best.
I read the Qudsī ḥadīth: "Whoever breaks his wuḍūʾ and does not perform ablution has shown me harshness; whoever performs ablution and does not pray has shown me harshness; whoever prays and does not supplicate to Me has shown me harshness; and whoever supplicates to Me and I do not answer him, then I have been harsh to him, and I am not a harsh Lord." What is the authenticity of this ḥadīth?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
This ḥadīth was mentioned by al-Ṣaghānī (may Allah have mercy on him) in al-Mawḍūʿāt (p. 43), where he said: “Among them is their saying: ‘Whoever breaks his wuḍūʾ and does not perform ablution has shown me harshness; whoever performs ablution and does not pray has shown me harshness; (whoever prays and does not supplicate to Me has shown me harshness); and whoever supplicates to Me and I do not answer him, then I have been harsh to him, and I am not a harsh Lord.’”
Al-ʿAjlūnī (may Allah have mercy on him) cited what al-Ṣaghānī said in Kashf al-Khafāʾ (Vol.2/p.264). And Allah Almighty knows best.
Is a woman sinful if she fasts out of shyness from her family while menstruating or in postnatal bleeding?
It is forbidden for a woman who is menstruating or in postnatal bleeding to abstain from food and drink with the intention of fasting. If she fasts out of shyness, she is sinful, as her fast is not valid.
However, if she refrains from eating and drinking without intending to fast, it is not forbidden, but she unnecessarily burdens herself without need.
Does tooth extraction during the day in Ramadan break the fast?
Simply extracting a tooth during the day in Ramadan does not break the fast. However, if water or blood enters the body cavity, the fast becomes invalid.
Whoever's fast is invalidated in this way must refrain from eating and drinking for the rest of the day out of respect for the sacred month and make up for that day later.
It is preferable to postpone the extraction until nighttime or after Ramadan if possible.