Is it permissible for one to give the Zakah (obligatory charity) to his indebted brother?
It is permissible for one to give the Zakah to his brother if he was indebted, or poor.
What is the ruling when the duration of the monthly period varies, and sporadically increases?
If the period didn`t exceed (15) days from the first seeing of blood discharges until its end including the times of temporary cessation, then it is menses. But, if the period exceeded (15) days, then the woman experiencing this is Mostahadah (bleeding outside the time of monthly period), and should abide by her regular period, and consult one of the scholars.
What is the ruling of Sharia on woman shaking hands with non-Mahram man?
All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
The Messenger of Allah (PBUH) said: "It is better to be stabbed in the head than touch a woman to whom one isn`t related." [Transmitted by Al-Tabarani /Al-Mo`jam Al-Kabir, Hadith No.16880]. Accordingly, it is forbidden for men to shake hands with non-Mahram* woman. And Allah The Almighty Knows Best.
* The women/men that you are allowed to marry. It includes all women/men other than mahram (including cousins).
What husband's stand should be, if his wife disagrees with his mother?
Both the mother and the wife have rights on you (Questioner); therefore, you are obliged to treat both justly, resort to wisdom to please both and intiate reestablishing cordial relations among them. And Allah Knows Best.