Is it permissible to kill sick animals out of mercy for them, and is it permissible to brand sheep on their faces with fire, and what is the reason for that?

It is not permissible to brand sheep on their faces, due to the prohibition of the Prophet, peace and blessings be upon him, regarding it, and his cursing of whoever does so. However, branding them on parts other than the face is permissible if it serves a valid purpose, such as distinguishing them. It is preferable for the branding to be on the ears of sheep and the base of the thighs of camels and cattle, as it causes less pain and makes the brand visible.