Is it permissible for a Muslim to employ a Christian servant to perform household chores such as cooking and cleaning, and what is the ruling of Islam on this matter?

It is permissible for a Muslim to employ a non-Muslim servant, subject to certain conditions, including:

The non-Muslim servant must not introduce impurity to the Muslim's food or clothing, nor cook anything forbidden to eat. Their employment must not lead to religious discord, such as admiration for Christianity, or Muslim children being influenced by rituals of disbelief, or the non-Muslim male or female servant becoming infatuated with the Muslim male or female employer, or vice versa. This must not be in the Arabian Peninsula, where it is forbidden for disbelievers to reside. The Muslim must not oppress them or withhold their rights.

It is desirable for a Muslim to strive to employ Muslims, as they are less prone to evil and further from committing forbidden acts. If the servant is not a disbeliever, the Muslim should invite them to Islam and strive for their guidance, following the example of the Prophet, peace and blessings be upon him, with the Jewish boy who served him.