Before I answer your specific questions let me begin with some background about the content and the meaning of the Kaddish.
The earliest references to the Kaddish that we have are in writings from the Mishnaic era. Tractacte Sofrim (approximately 200 C.E.) states:
"Mourners should say Kaddish after the Chazan finishes the Mussaf."
The earliest source for the actual text of the "Orphan's Kaddish" is in the works of the Gaonic period (approximately 600 - 900 C.E.).
In discussing the Kaddish, most commentaries quote an event recorded in various Midrashic texts as their model:
"Rabbi Akiva saw a strange and afflicted man who told him that he has been dead for many years and because he was particularly sinful in his lifetime he was judged harshly. When Rabbi Akiva taught the man's son to say Kaddish the man was spared the punishments."
This Midrash suggests that saying Kaddish is beneficial for the deceased. There is also a benefit for those who say the Kaddish. When a person suffers a loss they are often bitter and resentful towards God. They need the process of openly declaring His holiness and greatness to gradually bring them back to a recognition that the death is part of a greater plan and has cosmic meaning. The effect of this public declaration is also suggested as the reason for the benefit to the deceased. He has become the cause for a public sanctification of God's name which acts as an atonement for any desecration of God's name that his lifestyle might have caused.
- The most natural choice for those to say Kaddish for the deceased are his children. If the deceased has no living children, then his grandchildren children would say Kaddish. If he doesn't have grandchildren, then any relative can say if for him; and if he doesn't have any relatives, anyone can say it for him.
- Only his children are obligated to say it for him because of the commandment to honor one's parents. That is why when a parent-who had been saying Kaddish for someone-dies, Kaddish for this other relative ends. However, that doesn't mean that one can't say it on one's own initiative.
- May one say Kaddish for a non-Jew? Rabbi Oshry in his book, "Responsa from the Holocaust" writes: "In 1945 shortly after our liberation, Reb Moshe Segal came to me with the following question: He had been saved by a gentile woman who, at enormous risk to herself, had hidden him in her basement together with ten other Jews, providing them all with food and shelter until the liberation. After the war, when these Jews wanted to repay her in some way for her great compassion they discovered to their deep sorrow that she had died right after the liberation. The idea took root in their minds to say Kaddish for her, and Reb Moshe Segal was chosen for the task. His question was whether it is permissible to say Kaddish for a gentile?"
After mentioning a number of sources, Rabbi Oshry decided: "...it is clearly permissible to say Kaddish in memory of the gentile woman who saved so many Jews from death...May He who grants bounty to the Jewish people grant bounty to all the generous non-Jews who endangered themselves to save Jews."
- Tractate Sofrim, Chapter 19
- Gesher Hachayim, by Rabbi Yechiel Michel Tuchachinsky, Ch.30
- Responsa from the Holocaust, by Rabbi Ephraim Oshry, #85