Is killing one's wife on the pretext of infidelity and theft, without just cause, considered a sin, and does the killer fall under the threat of eternal damnation in Hell mentioned in the Quran, and can the wife remain married to him in the Hereafter after repentance and seeking forgiveness?

A Muslim man's protectiveness over his honor is a good trait, and a wife's infidelity via phone, if it's an emotional relationship with a stranger, is a reprehensible act. A woman must safeguard herself in her husband's absence, as Allah Almighty says: "So righteous women are devoutly obedient, guarding in [the husband's] absence what Allah would have them guard." And if she took money unjustly, that is not permissible for her.

However, this does not justify killing her, for killing her is a grave reprehensible act and one of the major sins, because it is killing without just cause. Allah Almighty has said: "And do not kill the soul which Allah has forbidden, except by right." You have done well by your remorse and repentance, and we hope that Allah will forgive you. The guardians of the victim's blood have the right to demand blood money (diyya), or they may pardon it if they wish, knowing that retribution (qisas) is not obligatory if a husband kills his wife and they have children together. However, if they have no children, then her guardians have the right to demand retribution.