Sanhedrin 19b
One who raises a child it is as if they gave birth to that child. From where do we learn this? From Ruth. "A child was born to Naomi. Did Naomi really give birth? Didn't Ruth? Yes Ruth gave birth but Naomi raised him, therefore he is considered as a child of Naomi.
Do we consider the child to be a 'real child' of the adoptive mother or father?
REMA Shulchan Aruch Choshen Mishpat 42:19
A person who raises an orphan in his home and writes on a contract 'my son' or the orphan wrote 'my father or mother', it is not considered fake, it is kosher since they raised him. (Rabbi Yaakov Emden--even if he is not an orphan)
How should they be called to the Torah? Ketubah? Get?
The Obligation to Honour
"At the time when God commanded Moses to go and save the Jewish people in Egypt, Moses responded 'I must first get permission from Yitro who is like my father'. One who raises someone and takes care of him deserves honour and respect more than a father or mother." Shmot Rabah
Chatam Sofer says obligation on the adopted son to honour the parents even when endangering one's life...
Whom to Adopt?
A Jewish child? Concerns about Mamzer!
If the child is a product of a Jewish married woman extra-marital affair, the child is a mamzer. If the child is a product of Jewish incestual relationship, the child is a mamzer. If the child's father is missing and the mother is quiet--shtuki and concern for Mamzer.
Thus ideally the parents are checked as to the status of their child and then no problems. If there is no proof Halacha says one may follow the majority of the city.
A Non-Jewish Child? Conversion!
גר קטן מטבילין אותו על דעת בית דין...הגדילו יכולין למחות וכו.
Is it a 'zechut' to adopt and convert if the family is not religious?
Yichud, Negiah
Lubavitcher Rebbe writes that is 100% prohibited! (Wozner, Flaxer, Kanyevsky...)
Commenting on the response of Rabbi Waldenberg,, I believe it is absolutely prohibited even with educational purposes, even when they adopted them from birth. The opposite is true, there is more desire among the step-relatives, thus prohibited is yichud, kissing, hugging and these are punishable by death!
However, Rabbi Eliezer Waldenberg (and to a certain extent Rav Moshe Feinstein, Rav Soloveitchik, Rav Rabinovich) were lenient
"It is clear that there is no room to be lenient even when the child doesn't know they are adopted...but after analyzing it carefully I can offer certain ways to be lenient so that your children will not have to separate from each other since they were adopted at an age where Yichud did not apply."
Rav Moshe also distinguishes between the date of adoption and permits when they were adopted at a very young age...
“And the daughter of Pharaoh came down to wash herself in the river” (Exodus 2:5), and Rabbi Yoḥanan said: She went down to wash and purify herself from the idols of her father’s house.
The Gemara understands that all the names referred to in the verse as children of Pharaoh’s daughter refer to Moses, as it will soon explain. The Gemara asks: Pharaoh’s daughter bore Moses? But didn’t she merely raise him? Rather, it is telling you that with regard to anyone who raises an orphan boy or girl in his house, the verse ascribes him credit as if he gave birth to him.