Is it incumbent on a family to provide for its old handicapped son and take care of him?
The family and relatives- e.g.brothers- of such a person are obliged to spend on him and take care of him if he didn`t have money of his own.
Is it permissible to shorten the eyebrows if they get too long?
Yes, it is permissible, but not desirable.
Is it permissible for me to name my newborn baby boy “Al-Moustafa” ?
There is no harm in naming your son “Moustafa” without using the ”Al”.
Is it permissible to single out Fridays, Saturdays, and Sundays to make up for missed fasting without fasting a day before ,or after each ?
Yes, it is permissible to make up for missed fasting during the aforesaid days, and it is also valid to offer non-obligatory and vowed fasting as well. However, fasting a day before or after each isn`t obligatory. And Allah Knows Best.