Is Aqeeqah due on the miscarried embryo?
No Aqeeqah is due on the dead embryo.
Is it permissible for a woman to sit with her brothers –in- law?
It is impermissible for a woman to sit with her brothers-in-law, and it is also impermissible for a person to have a Khulwa(seclusion) with his sister-in-law.
Is Iddah (waiting period) incumbent on an old lady whose husband had died?
Upon the death of the husband, Iddah is incumbent on the wife regardless of her age.
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.