Is it permissible for a woman to shake hands with her brother-in-law?
It is forbidden for the woman to shake hands with her brother-in-law because he isn`t a Mahram (unmarriageable) of hers.
What is the ruling on performing istinja' after passing wind from the anus?
Ablution is invalidated by passing wind from the anus, even without a smell. Only ablution becomes obligatory from it. Istinja' is not obligatory nor recommended. And Allah the Almighty knows best.
What is the ruling on discharges two or three days before the expected period of menstruation and the light-colored blood accompanying it? Must the woman abstain from prayer during this period?
If the total duration of these blood-tinged discharges reaches a day and a night (24 hours) or more, it is considered menstruation. If their total does not exceed 24 hours, it is not considered menstruation but is irregular bleeding (dam al-fasad). Each woman has specific cases for which she should ask the scholars. And Allah the Almighty knows best.
Is it incumbent on the fiancée to obey her fiancé?
When the woman settles in her husband`s house, it is incumbent on him to provide for her and it is incumbent on her to obey him. Before that, and if the marriage contract had been concluded, then she is lawfully his wife and thus she should abide by custom in treating him, but if the marriage contract hadn`t been concluded, then she should treat him as a non-Mahram (Marriageable).