Is it permissible to shorten the eyebrows if they get too long?
Yes, it is permissible, but not desirable.
A man has married a second wife and deprived the first from provision and overnight stay. He spent most of his time and wealth on his second wife. After sometime, the latter got sick and was diagnosed with breast cancer, which made him forsake her in bed and return to his first wife. What is the position of Sharia on this?
All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon Prophet Mohammad and upon all his family and companions.
It is forbidden for a woman to forsake her husband in bed without a lawful excuse since this is one of the mutual rights of spouses. When the wife abstains from making love to her husband without a lawful excuse, she is considered sinful. This is because the Prophet (PBUH) said: "If a woman spends the night deserting her husband's bed (does not sleep with him), then the angels send their curses on her till she comes back (to her husband)." [Agreed upon]. And Allah The Almighty Knows Best.
Is it permissible for a Muslim to escort his deceased disbelieving relative to his final resting place?
It is permissible provided that he doesn`t walk behind non-Islamic symbols, rather he should walk in front of them.
Is it permissible to offer an absentee funeral prayer?
Yes, it is permissible to offer an absentee funeral prayer.