Is it permissible for a Muslim woman to wear trousers at home?
Wearing trousers at home isn't prohibited so long as non-Mahrams do not see her. 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.
Does the clipping of nails nullify ablution?
No, it does not, and it is preferable to wash the hands afterwards.
Is it permissible to offer an absentee funeral prayer?
Yes, it is permissible to offer an absentee funeral prayer.