What is the ruling on singing and listening to it while accompanied and unaccompanied by musical instruments?

Singing and listening to it unaccompanied by musical instruments is permissible as long as it isn`t practiced as a profession and is done by a man to other men, or a woman to other women with maintaining that no temptations are involved. The words of the song should be Islamic, and if temptation was likely to happen, then the singing becomes definitely forbidden. Temptation here means: whatever leads to committing a sin. In Islamic Sharia, singing accompanied by musical instruments such as the lute and the like is forbidden, and listening to it as well. The reason behind that is the fact that the musical instrument itself is forbidden. However, striking the tambourine and singing Islamic songs are permissible in weddings, circumcisions and the like.

Is it permissible for a person to sever ties with his sister because her husband had quit prayer and done him harm?

A sister is a blood-relative and maintaining blood-relations is mandatory in Islam. As regards her husband, he should be invited to the way of Allah with wisdom and beautiful preaching.

If the Prophet (PBUH) was an Arab citizen in 2009, born in an Arab country ruled by an Arab ruler, saw what was happening to Gaza, what would he (PBUH) have done?

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.