What is the ruling on a Muslim participating in the slaughter of a sheep with a non-Muslim, and is it forbidden for him to assist in skinning, cutting, and transporting it? And what should he do if refraining from that will expose him to problems?

If the slaughterer is a person of the Book (Christian or Jew), there is no objection to assisting him in skinning and cutting the meat, because the slaughtered animals of the People of the Book are permissible. However, if the slaughterer is a non- الكتابي (non-People of the Book) disbeliever, his slaughtered animal is not permissible. You must be the one to slaughter for the dhakah (lawful slaughter) to be valid; if a non-Kitābī disbeliever performs the slaughter directly, the dhakah is not valid.

For a valid dhakah, it is a condition that the slaughterer be a Muslim or a person of the Book. The slaughtered animal of a non-Muslim and non-Kitābī, such as a Majūsi (Zoroastrian), an idol worshipper, or an apostate, is forbidden. If a Muslim participates in the slaughter with a non-Kitābī, then the slaughtered animal becomes forbidden. What is slaughtered by a non-Kitābī is considered carrion (maytah) and impure, and participating in its skinning and cutting involves touching impurity. There is a difference of opinion regarding the ruling on touching impurity without need.