Is it permissible, in Sharia, not to slaughter the Udhiyah (sacrificial animal) and instead donate its value to the needy—including parents and relatives—due to the authorities requiring that the sheep be sedated before slaughter, or that the slaughter take place far from the city without the presence of the Udhiyah's owner?

The Udhiyah (sacrifice) is a recommended Sunnah, not an obligation. It is preferable for the owner to slaughter it himself. If this is not possible, then it is better to appoint a Muslim to slaughter it. It is not advisable to appoint a non-Muslim, due to scholarly disagreement regarding the validity of such an act. If slaughtering or appointing someone to slaughter is not possible in the sacrificer's country, then it is permissible to appoint someone in another country, such as the country of her parents. It is a Sunnah to give away parts of the Udhiyah as gifts and charity. If she does not appoint someone for the sacrifice, it is permissible for her to donate its value to whomever she wishes.