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 for the father to give the Zakat of his money to his poor married son?
It is permissible for the father to give part of the Zakat to his poor son, if the latter was sound and healthy, because in this case the father isn't obliged to spend on his young son, even though; he is considered one of the Muslims poor. And Allah Knows Best.
I have vowed that if Allah gave me what I wanted so bad, I would fast every Monday and Thursday to the rest of my life; unless there is a valid excuse hindering me from doing so. Fortunately, Allah gave me what I wanted, so I started fasting Mondays and Thursdays; however, I stopped out of laziness. What is the Sharia ruling on this?
All perfect praise be to Allah the Lord of the Worlds. May Allah`s peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
Allah The Almighty described the righteous people of Jannah where He Said (What means): "They perform (their) vows, and they fear a Day whose evil flies far and wide." [Al-Insan/7]. Accordingly, you must do your best to fulfill your vow. However, if you are unable to do so, then feed a poor person for every missed day of fast (About half a JD), and if you are unable to do that, then offer expiation for breaking a vow, which is feeding ten poor Muslims (About half a JD for each), ask Allah for forgiveness and increase remembrance of Him. And Allah The Almighty Knows Best.
Is Making up Missed Obligatory Prayers an Obligation?
All perfect praise be to Allah, The Lord of The Worlds, and may His peace and blessings be upon our Prophet Muhammad and upon all of his family and companions.
Missed obligatory prayer/prayers is a debt upon Muslim and making it up is an obligation, since the Prophet (PBUH) said: "pay the debt due to God, for it is the one which most deserves to be paid." [Agreed upon]. And Allah Knows Best.