1) It is extremely important to recite kaddish for the deceased. It is clear from the words of the Zohar that Kaddish helps the soul from the punishments of Gehinnom. The Ari z”l (Shaar Hakavanos Kaddish) adds that tt is also beneficial in raising the deceased’s soul from one level to another in Gan Eden. Jews therefore customarily recite Kaddish following the passing of a father or mother, even if they were exceptionally righteous individuals.
2) The Meforshim tell us that by reciting Kaddish for one’s parents who are deceased one fulfills the mitzvah of Kibud Av V’Em. (Derech Pekudecha 33 and Rav Yosef Engel in Gilyonei Hashas 31b)
3) The Chasam Sofer (E.H. 69) writes that he has heard that one does not recite Kaddish for Jews who were not observant and who did not adhere to the Torah. He explains that this is unequivocally false and that on the contrary these souls need Kaddish much more than anyone else. This is the accepted approach amongst the poskim (for a larger discussion see Umekareiv Biyamin vol. 1 page 275).
4) The poskim discuss whether a convert can recite kaddish for his non-Jewish parents. Harav Ovadia Yosef zt”l (Yechava Daas 6:36) writes that one is absolutely permitted to do so. Harav Ahron Walkin zt”l (Zaken Ahron 2:87) writes that they are permitted to do so. But, he adds if there are people in the shul that are aware that he is reciting it for a non-Jew it may seem strange and in that instance it is better to learn Mishnayos than to recite Kaddish. However, Harav Yaakov Hillel shlit”a (Vayeshev Hayam 3:32) maintains that one should not recite kaddish for a non-Jew. He explains that the convert can privately (in his shemoneh esrei) pray to Hashem to aleviate the suffering of the ould of the departed etc. For normative halacha, a rav should be consulted.
5) The custom is to say Kaddish for only eleven months so as not to suggest that one’s parent was an evil person (rasha). This is because the evil are judged for twelve months. (Rama Y.D. 376:4)
6) The poskim discuss the following case: If a person was not observant and did not recite kaddish during the eleven months of the mourning and years later became observant can they then recite kaddish for eleven months or at that point it would not be permitted. Harav Kook zt”l (Daas Kohen 217) rules that it is forbidden, he explains that if reciting kaddish for twelve months is not allowed so as not to suggest that one’s parent is evil. All the more so that one cannot begin reciting kaddish years later. However, Harav Yaakov Hillel shlit”a, for reasons beyond the scope of this halacha, rules that one may recite kaddish even years later (see Umekareiv Biyamin 2:1). For normative halacha, a rav should be consulted.