Laws of Kibud Av Vaeim relevant to children who were adopted:
A. Is an adopted child obligated in the laws of Kibud Av Vaeim towards his adoptive parents?
A child who was adopted is not obligated in the scriptural Mitzvah of honoring and fearing his adoptive father and mother. Nonetheless, he is to show them respect more than the norm, as although he is not obligated in the mitzvah of Kibud Av Vaeim, he is obligated to show them gratitude for bringing him up and troubling themselves on his behalf. [In this regard, it is similar to a convert who is to still show some extra respect to his biological parents despite not being obligated in the scriptural laws of Kibud Av Vaeim. Likewise, he may be obligated to honor them similar to the obligation for one to honor his Rebbe. Thus, for example, although he is not obligated to stand up for them when they enter a room, the adopted child should try to care for them and serve them their requests.]
May one say Kaddish for the passing of an adopted parent? Yes, an adopted son is to say Kaddish on behalf of his adoptive parents after their passing if there are no biological sons saying Kaddish for the deceased parent.
B. Is an adopted child obligated in the laws of Kibud Av Vaeim towards his biological parents?
Yes. He is obligated in all the laws of honoring and fearing his biological parents just as any other child would be obligated towards his parents. This applies even though his biological parents have never done any kindness towards him, and perhaps have created resentment in the child for having given him over for adoption, nonetheless the biological child is still obligated in the mitzvah.
Should an adopted child place effort to discover the identity of his biological parents so he can honor them and fulfill the mitzvah of Kibud Av Vaeim? This matters debated amongst today’s Poskim. While some say that it is proper to discover their identity in order to fulfill the mitzvah, others say that it is better not to in order not to put oneself into a challenge of transgressing this mitzvah. [Practically, the adopted child should weigh the pros versus the cons of knowing the identification of his biological parents before making such a decision and seek consultation if necessary.]
 See Nishmas Avraham 4:39
 Chasam Sofer O.C. 164; Lechem Shleima Y.D. 2:72; Mishneh Halachos 4:70; Ashreiy Haish 40:16 in name of Rav Elyashiv; Pesakim Uteshuvos 240:56; See; Megillah 13a; Sanhedrin 19b; Shemos Raba 4:2
 Chasam Sofer ibid based on Sotah 49a and Rashi there regarding Rav Acha Bar Yaakov whose grandson whom he also adopted stated that he does not have to bring him a cup of water being that he is not his biological son;
 See Nishmas Avraham ibid in name of Rav SZ”A that for all practical purposes, one should respect them in the same manner that one would respect his biological parents
 In truth, the custom is to be lenient in this matter even regarding biological parents under the assumption that they forgive their honor, although by biological parents we initially rule that one should explicitly receive forgiveness from them regarding standing up in their honor. This forgiveness would not be necessary to be received from adoptive parents.
 Chasam Sofer O.C. 164 and Y.D. 345; Nitei Gavriel 49:8; Nishmas Avraham ibid; See Sdei Chemed Asifas Dinim Aveilus 156
 See Chinuch Mitzvah 33; Michaber 240:8 and 18; Meshech Chochmah Devarim 5:16; Igros Moshe Y.D. 2:130; Nishmas Avraham ibid
 See Vavei Hamudim Vechashukeihem p. 28
 See Brachos 53b that one is obligated to chase after a mitzvah, brought in Minchas Yitzchak 2:75; Menachos 41a that in a time of wrath one is punished even for not fulfilling a non-obligatory positive command
 See Kiddushin 31b that due to its severity and demanding details, Rebbe Yochanon stated that from a certain perspective, one who has not met his parents benefits from the fact that he will not be subject to being punished for not properly honoring them.; Conclusion of Rav Chaim Kanievsy and Rav Zilbishtrom ibid; A similar argument has been brought in Poskim regarding not moving to Israel being that it is difficult to follow all the laws dependent on the land