1. Can we distribute all of the meat of the offerings to the poor, not keeping anything for ourselves and relatives? I mean is it alright if all of the meat is distributed to those who deserve irrespective of if they are relatives or strangers? 2. Can we distribute the meat of the sacrifice to non-Muslims and non-believers who are poor and deserving? 3. Is there any religious ruling that we have to give preference to Muslims over non-Muslims when distributing the meat? 4. If there are two persons equally deserving, one is Muslim and other is non Muslim, who should we give meat to? Please provide answers for all questions separately.
Almost all your questions can be answered based on the verse 22:37 of the Qur'an about the meet of a sacrificed animal:
(So eat from it your selves and also feed those who are content and those who ask.
فَكُلُوا مِنْهَا وَأَطْعِمُوا الْقَانِعَ الْمُعْتَرَّ'
1. Since, the above verse has a permitting style and not an instructing style, to eat or not to eat from the meet of a sacrificed animal is optional.
2. There is nothing in the above verse to restrict the consumers of the meet to Muslims and believers therefore there is no basis to say it cannot be given to non-Muslims or non-believers.
3. As explained above, the answer is no.
4. It is difficult to assume a situation where two people are absolutely deserving a food. Every individual has his/her on very specific circumstances and might have dependants with very different conditions. I therefore think that the situation illustrated by the Question never materializes. However to answer the question from theoretical point of view, again there is nothing in Shari'ah to tell us that in such situation the Muslim should be preferred. However, it is very logical and appropriate if a Muslim prefers a fellow Muslim in such situation.
Questions 2, 3 and 4 are answered with the assumption that the non-Muslim under discussion is not personally in the state of war with Muslims.