Q. Can a convert sit shiva and recite kadish for his benevolent and righteous father or mother that did not convert? How about if they were actively keeping another religion?

A. In question 824 we wrote; The Talmud (Yevamos 22a) teaches that although a ger who converts is considered as a new born person and looses his familial connections to his biological family, rabbinically he cannot marry his sister or other close relatives. The Talmud explains that the reason is, people will say the ger left a stronger sanctity and converted to a lesser and more lenient religion. Rambam (H. Ma’amrim 5: 11) rules that not only should a ger observe the negative commandments in regards to dishonouring or shaming his parents, but he should also show some honor and respect to them. (See also Shulchan Aruch Y.D. 241: 9) Yad Avrohom explains that honoring parents is an obligation for Bnei Noach, as we can infer from the narrative of Chom Ben Noach, yet they are not punished for non-compliance as they are in the case of the other seven Noahide mitzvos. Rabi Akiva Eiger in his notes (ibid.) maintains that it is referring to the way Gentiles in their own beliefs and laws, greatly honor their progenitors. Igrois Moshe (Y.D. 2: 130) in a similar shaile rules that even when the parent is not ill, the ger is permitted an occasional visit and is also allowed to bring along his children. However, he should do so infrequently so that the children will not adhere to the former religion of their father, join his grandparents in eating non-kosher and do other common prohibitions. Otherwise, in case of need or illness, just as one has to take care and visit the afflicted gentiles and tend to their needs, certainly and more so, the ger has to take care of his own parents. (See also Nishmas Avrohom 2: p. 276) Horav Shlomo Miller’s Shlit”a opinion is similar. (See also question 825 in this forum).
Yechave Daas (6: 60) rules that a convert can recite kadish for his parents, say the customary hashkava and pray leilui nishmosom and elevation of their souls. Yalkut Yosef (Avelus H. Kadish 12) adds that he should recite kadish in an unpretentious and demure style, not to rise the criticism of others. (See also Miyam Hahalacha 2: 16, Eitz Hachaim p. 305).
Shulchan Aruch Y.D. 374: 5) rules that a convert does not sit shiva for his parents even if they also converted, since they are regarded as being reborn again and to have lost their filial connection. (Be’er Hagola ibid. 10). Some maintain that according to the Poskim that first day avelus is Biblical, they have to sit that one day. However, the Halacha does not follow that opinion (Shach ibid. 4).
Horav Shlomo Miller’s Shlit’a opinion is that a convert should not sit shiva or recite kadish for his parents, but should offer his sincere and heartfelt prayers to Hashem in their behalf.
Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit’a