Does nosebleed during the day in Ramadan affect the validity of fasting?
A nosebleed does not affect the validity of fasting unless some of the blood reaches the body cavity or the person intentionally swallows it. In such a case, their fast is invalid, and they must refrain from eating and drinking for the rest of the day and make up for that day later.
Does Laylat al-Qadr move between the nights of Ramadan?
Laylat al-Qadr is definitely in Ramadan, and it is most likely in the last ten nights, especially on the odd-numbered nights.
For this reason, the Prophet ﷺ commanded observing worship in the last ten nights, and he himself used to dedicate those nights to worship. Many people put special effort into the 27th night of Ramadan, and they are rewarded in any case, whether they actually coincide with Laylat al-Qadr or not, because worshiping on any night of Ramadan carries great rewards.
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.
What is the ruling on a suitor (khāṭib) going out alone with his fiancée before the marriage contract is concluded?
It is unlawful for a suitor, prior to the marriage contract , to do anything that would be unlawful for any non-maḥram man — including touching his fiancée, being secluded with her (khalwah), or seeing her without her complete sharī‘ah-compliant covering — even after the recitation of al-Fātiḥah (This is a cultural custom to seek Allah's blessing (barakah) and mark mutual consent) and the family's approval [of the engagement]. This is because the engagement, without the formal marriage contract, does not render unlawful matters permissible. And Allah, the Exalted, knows best.