He found a baby and adopted him – what is the ruling?

Praise be to Allah.

Firstly: 

Your father and mother are to be thanked for what they did, taking care of this foundling until he grew up. 

Secondly: 

Radaa’ah or the breastfeeding that makes the child a mahram means giving five or more feedings within the first two years of the child’s life. If your mother breastfed this child in this manner, then he becomes a son for her and her husband through breastfeeding, and he becomes a brother to all their children through breastfeeding. Allaah says (interpretation of the meaning): 

“Forbidden to you (for marriage) are: your mothers, your daughters… your foster milk suckling sisters”

[al-Nisa’ 4:23] 

“The mothers shall give suck to their children for two whole years, (that is) for those (parents) who desire to complete the term of suckling”

[al-Baqarah 2:233]

And it was proven that the Prophet (peace and blessings of Allaah be upon him) said: “Relations through radaa’ah (breastfeeding) makes all those things unlawful which are unlawful through corresponding birth (blood) relations (i.e., prohibited marriages).” Narrated by Maalik, 2/601; a similar version was narrated by al-Bukhaari, 3/149; Muslim, 2/1086. And it was narrated that ‘Aa’ishah (may Allaah be pleased with her) said: “One of the things that was revealed in the Qur’aan was that ten recognized breastfeedings makes all those things unlawful which are unlawful through corresponding birth (blood) relations, then that was abrogated and replaced with a verse that mentioned five, and the Messenger of Allaah (peace and blessings of Allaah be upon him) passed away when that was the ruling.” Narrated by Maalik, 2/608; Muslim, 2/1075 

It should be noted that what is considered to be one breastfeeding is when the child takes hold of the nipple and sucks milk from it; if he lets go, then comes back and sucks more milk, this is a second breastfeeding, and so on. 

Thirdly: 

It is not permissible for the child mentioned to take your father’s name as if he is one of his sons. 

Fourthly: 

The child mentioned cannot inherit from your father, because he is not one of his heirs. 

Fifthly: 

If it is proven that your father left one-third or less of his estate to the child mentioned, there is nothing wrong with that; you may also keep in touch with him and treat him kindly, for Allaah will never cause the reward of those who do good to be lost. 

And Allaah is the Source of strength.