The Gemara Sanhedrin 19b says that anyone who raises a child it is as though he gave birth to him. Rav Soloveitchik (quoted in Chavrusa Nissan 5742) suggested that raising someone else’s child is a partial fulfillment of the mitzvah of Pru Urevu (reproduction) and bases it on the Rambam (Ishut 15:3). See Chachmat Shlomo EH 1:1 who agrees with this approach.
- The child raised by someone else has a strong moral obligation to respect his adopted parents possibly with even greater than the respect displayed by children towards their biological parents. 
- An adopted child should say Kaddish for his adopted father. 
- The Lubavitcher Rebbe (quoted in Otzer HaPoskim EH vol 9, pg 130 and 263) raised the issue of Yichud and Negiyah between a boy and his adopted mother. Dvar Yehoshua 3:16 and Chazon Ish (quoted by Peni HaDor vol 3 pg 40) were strict on this issue. However, Sh”t Tzitz Eliezer and Rav Soloveitchik quoted by Rav Melech Schachter in RJJ Journal (vol 4, pg 96) were lenient on these issues.
- Rav Melech Schachter in RJJ Journal (vol 4, pg 94) based on Shemot Rabba chapter 4
- Rav Melech Schachter in RJJ Journal (vol 4, pg 95) based on Beiur Halacha (end of 132) and Sdei Chemed (Avielut 156). See also Chatom Sofer OC 164.