Is it permissible to offer the Eid sacrifice in a non-Muslim country?
Eid sacrifice is permissible anywhere, and Islamic Sharia hasn`t determined that it should be in an Islamic country, but poor Muslims should be given a share in order for it to be valid since it is an act of worship. A portion of that Eid sacrifice should be given at least to one poor Muslim, and if that wasn`t possible then the one offering it should consume it along with his family.
Is it permissible for one to shake hands with his uncle`s daughter?
No, it isn`t because he is a non-Mahram (Marriageable) to her.
What is the ruling on smoking?
All Perfect Praise is due to Allah, The Lord of The Worlds
Smoking is forbidden (Haram) since it is harmful to one`s health as confirmed by trustful physicians, and Islam has forbidden taking anything that causes harm. The Prophet (PBUH) said: "There should be neither harm nor reciprocated harm." [Ibn Majah and Ahmad]. And Allah Knows Best.
What is the ruling of Sharia on tubal ligation?
All perfect praise be to Allah, the Lord of the Worlds; may His blessings and peace be upon our Prophet Mohammad and upon all his family and companions.
Undergoing tubal ligation* to protect the mother`s health is permissible provided that this procedure is performed based on professional opinion of trustworthy medical doctors. And Allah The Almighty Knows Best.
*Tubal ligation is a surgical procedure for female sterilization in which the fallopian tubes are permanently blocked, clipped or removed. This prevents the fertilization of eggs by sperm and thus the implantation of a fertilized egg. Tubal ligation is considered a permanent method of sterilization and birth control.