Do half-siblings inherit the blood of the deceased and have the right to claim and relinquish it, or is this right limited to full-siblings and the mother of the deceased?

The madhhab of the majority of scholars is that the awliya’ al-dam (those entitled to retribution for bloodshed) are the heirs of the deceased, both men and women. They have the right to qisas (retribution in kind) or to waive it, either for a blood-money (diyah) or without it. There are narrations in the Sunnah indicating that women may pardon bloodshed, and that the term 'wali' includes female heirs, even if the kinship is through marriage.