Is it permissible to distribute the raw meat of the vowed animal sacrifice amongst the poor/to offer it to them in cooked form?

It is impermissible for the vow-maker to eat from the vowed animal sacrifice, rather, he/she should distribute it as he/she had intended upon making the vow, but if the vow was a general one, without any specification then, it is better to give it as raw meat.

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.

A pious man proposed to me and my father was hesitant in this regard because he is black, am I sinful if I accepted his proposal?

If the suitor is pious, color isn`t a drawback. However, try talking to your father kindly, so that he approves of your marriage with contentment, and that is better for you.

What is the ruling on working for orphans' institution?