Is it permissible for me to continue raising and providing an Islamic upbringing for my 6-year-old child, knowing that I intend to marry his mother, and is this child considered a "child of the bed" (ibn firash) whose lineage cannot be attributed to me?

The majority of jurists hold the view that a child born of adultery is not attributed to the fornicator, even if he acknowledges paternity, and this is the fatwa we adopt. There is another opinion by Ibn Taymiyyah and his student Ibn al-Qayyim allowing such attribution. If adopting this opinion averts a major harm, then there is no objection. As for your marriage to this woman, if she is a Muslim or a woman of the People of the Book and has repented sincerely from adultery, then it is permissible for you to marry her, even though we do not prefer marriage to a woman of the People of the Book.