Is it permissible for one to shake hands with his uncle`s daughter?
No, it isn`t because he is a non-Mahram (Marriageable) to her.
What is the ruling on discharges that occur two or three days before the menstrual period, accompanied by a light blood color? Should I stop praying during this period?
Brownish, yellowish, and reddish discharges are considered part of menstruation if they occur within the menstrual period, provided that the total duration does not exceed fifteen days from the first discharge to the end of the cycle; in this case, all of it is considered menstruation.
What is the ruling on buying a car from someone for less than the market price, because they are in a hurry to travel and want to sell their belongings at a lower price due to their urgency?
Yes it is permissible; however, a Muslim shouldn't take advantage of other's difficult situations.
Should vowed fast be offered in consecutive days, or not?
Vowed fast should be offered in line with what was originally intended by the vow-maker, either consecutively, or not.