Is a wife considered divorced if her husband refused to go to bed with her (for sexual intercourse) for two months?
A wife doesn`t get divorced no matter for how long her husband refuses to have sex with her, and she had better go to court.
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?!
Can someone who begins a voluntary fast break it?
It is preferable for someone who begins an act of worship not to break it.
Allah the Exalted has said {what means}: "and let not your [good] deeds come to nought!" [Muhammad/33].
However, if a person starts a voluntary fast (nafl) and needs to break it, they are going against what is preferable, but there is no sin upon them.
How many rak‘ahs are there in Tarawih prayer?
The most complete form of Tarawih prayer consists of twenty rak‘ahs, excluding Shaf‘ and Witr. This is the opinion of the majority of scholars, both past and present, and it is the practiced tradition in the two Holy Mosques.
However, Tarawih is valid with any even number of rak‘ahs, even two rak‘ahs, as long as it is performed with the intention of Qiyam al-Ramadan.