Is it forbidden to adopt a foundling, even though he is blameless, knowing that an orphan is not registered under the adopter's name nor does he inherit from his wealth?

Adoption is absolutely forbidden, whether the adopted child is a foundling, an orphan, or otherwise, due to the generality of the prohibition.

As for the care and upbringing of a child, it is an act of righteousness and kindness. This ruling applies equally to a foundling and others. However, it is not permissible to attribute him to the one who adopted him, nor do the rulings of biological sonship, such as inheritance, apply to his care and upbringing. It is permissible to bequeath to him up to one-third of the wealth or less.