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.
Who is "the poor" entitled to receive Zakah (obligatory charity)?
The poor is the one who has neither money nor a source of living, or has either of them, but it isn`t sufficient such as being in need for a hundred/JDs, and having an income of forty/JDs only.
What do you say about someone who stays up all night on Laylat al-Qadr but misses Fajr prayer?
Such a person has deprived themselves of a great reward. Performing obligatory acts is more beloved to Allah than voluntary acts. Fajr prayer is an obligation, and performing it in congregation is equivalent to worshiping for the entire night. So how can someone neglect it while striving for voluntary prayers?!
What is Aqeeqah?
It is the sheep slaughtered on the seventh day from the child`s birth, and it is a confirmed Sunnah after the Prophet (PBUH).