What is the ruling on marrying a Muslim woman who is a recent convert and has a child from a previous relationship, and will this child be a mahram to the questioner's sister and mother if they live together, and is it permissible for the questioner to adopt this child?

It is permissible to marry a woman who bore a child out of wedlock if she repents and has purified herself. If the child is of nursing age, he becomes a mahram to your mother and sister if he nurses from a woman who establishes mahramiya for them, or simply by nursing from his own mother after your marriage to her, according to the Maliki school. However, the majority of scholars do not hold this view. If he has passed the age of nursing, mahramiya is not established between him and your mother and sister through your marriage to his mother. It is permissible for him to live with them in the same house as long as he is at an age where lust is not feared. It is not forbidden to look at his awrah or touch it before the age of seven, provided lust is not feared. Upon reaching the age of lust, it is not permissible for him to live with them unless there are provisions to prevent khalwa (private seclusion). As for registering the child in your name, relevant fatwas can be consulted.