Joseph,
No, a Kohen is not by default on a higher spiritual level than a Yisroel. The priority of a Kohen before a Yisroel is because of the kavod that Hashem requires we give the Kehuna. So all things being equal we give kavod to the Kohen but if we know that the Yisroel is a greater talmud chochom, the Yisroel comes first because the kavod HaTorah comes first.
That being said, I don’t think you are correct that we necessarily save the life of a Kohen first. The Mishna in Horios that I think you are referring to, is talking about kedima in tzedaka or kavod. See Shulchan Aruch, Yoreh Deah 351:9.
In the case of a woman and a man, when it comes to tzedakah the Shulchan Aruch paskens that the woman has kadimah. (Yoreh Deah 351:8) When it comes to hatzalas nefoshos, the Rema paskens that the man comes first (Yoreh Deah 352:8). The meforshim explain that this is because a man has more mitzvos than a woman (i.e. the greater potential of the man wins out over the higher floor of the woman). But if one knew that the woman was righteous and the man was not, b’pashtus the woman would come first. See Yerushalmi Horiyos 3:4.