Is it permissible to pay the Fitr Zakah (obligatory charity) of Ramadan on behalf of a dead person?

The Fitr Zakah of Ramadhaan isn`t due on one who had passed away before the sunset of the last day of Ramadan. And Allah Knows Best.

Should I pick up the newspapers thrown at the street because they definitely contain sacred names?

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) says: "Be mindful of Allah and He will protect you." Part of this is protecting whatever contains the name of Allah. Therefore, if you see a paper containing Allah's name thrown in the street, then you should grab it and similar papers, then burn them in a clean place. Doing so is permissible because Othman (May Allah be pleased with him) has burnt copies of the Quran which didn`t match the main Mus-haf. However, you aren`t required to collect every paper in the street. And Allah The Almighty Knows Best.

Is it permissible to make up for missed fast after the beginning of the second half of the month of Sha`ban (the month before Ramadan)?

Yes, it is permissible, but one who had missed fasts should hasten to make up for them. As for the Hadith mentioned in this regard, the prohibition is for offering absolute voluntary fasting. And Allah Knows Best.

After our father had passed away, my brother and I stayed with our mother for twenty years and served her to the best of our effort. However, she asked our other brothers (3) to support her financially but they refused to pay her anything. As a result, she gave her share of the inheritance to me and my brother. What is the ruling of Sharia on this?