What is the ruling on purchasing sacrificial animals and authorizing their slaughter via telephone?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is preferable for the one offering a sacrifice (udhiyah) to slaughter the animal himself, in adherence to the Sunnah, or to be present at the slaughter if possible.
It is valid to authorize a butcher to purchase a sacrificial animal on his behalf and slaughter it, regardless of whether the animal is owned by the butcher or the butcher is acting as an agent in its sale—according to the Hanbali school—provided that the butcher specifically designates and purchases the animal for the person offering the sacrifice before slaughtering it. And Allah Almighty knows best.
Is it permissible to offer mandatory prayer while sitting?
All perfect praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions. Standing up is a pillar in the mandatory prayer and the prayer is not valid without it unless one cannot do it. As for Nafila (supererogatory prayer), one can perform it in the state of sitting even if he/she can stand, but his/her reward is half the reward of the person who prays while standing if he/sh has no legal excuse for such an act. And Allah Knows Best.
Is it permissible for one who sacrifices on behalf of another, with the latter's permission, to eat from the sacrifice?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is permissible for one who sacrifices on behalf of another with his permission to eat from the sacrifice with his permission, and he takes his place in distributing its meat.
It is stated in Hashiyat al-Jamal 'ala Sharh al-Minhaj (Vol.5/P.262): 'If a person sacrifices on behalf of a living individual with his permission, does he take the place of that individual in distributing the meat, because permission for the sacrifice implies permission for distribution, or does it depend on explicit permission? There is room for consideration, but the former view is not far-fetched.' And Allah Almighty knows best."
Is it permissible for the young woman to turn the suitor down if she had no feelings for him?
Yes, it is permissible for her to do so.