Does using a gargling medicine break the fast?
If the medicine reaches the body cavity (jauf), the fast is invalidated. However, if it does not enter the body cavity, the fast remains valid.
Therefore, it is advisable to avoid using it during the day in Ramadan.
What is the ruling of Islamic Law on participating in the prize draws (raffles) that commercial stores hold for their customers?
Praise be to Allah, and peace and blessings upon our master, the Messenger of Allah.
There is no objection to participating in the prize draws (raffles) that commercial stores hold for their customers, provided that the buyer's intention in purchasing is not merely to enter the competition without any need for what is bought — rather, the purchase must be genuinely intended for the item itself, and one must not pay more than the item's fair market price. This is because paying an amount above the market price would effectively be paying a fee to enter the prize draw, which would render it a form of gambling (qimār).
These prizes are, in essence, gifts that businesses offer through a random drawing (qur'ah) to those who purchase from them, as a means of encouraging sales, without the customer bearing any additional monetary cost for participation. So long as the aforementioned conditions are met, there is no objection to benefiting from the prize offered by the store, as it is considered a lawful prize from the viewpoint of Islamic Law.
It is stated in the resolutions of the "Jordanian Iftaa' Board" (Resolution No. 47), in the context of outlining the conditions for permissible prizes: "The price of the ticket [or item purchased] for the sake of the prize must not exceed its original price, so that there is no payment of money in exchange for participation in the draw." And Allah, the Most High, knows best.
Does the clipping of nails nullify ablution?
No, it does not, and it is preferable to wash the hands afterwards.
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.