ما هي واجباتي تجاه طفل زوجتي، وهل يجوز لي أن أنسبه لاسمي أو أن يناديني بـ "أبي" مع وجود والده الحي؟

You are rewarded for caring for and sponsoring this child. As for adoption and attributing him to yourself, this is not permissible in Islamic law, for Allah Almighty says: "Call them by [the names of] their fathers; it is more just in the sight of Allah." Muslims have unanimously agreed on the impermissibility of adoption and attributing a child to other than his father.

As for the child calling you "O my father," there is no harm in it if it is for the purpose of reverence and respect, and because he sees you as being in the position of his father. However, it is preferable to avoid habitually using it to prevent any misconception.