What are the rulings concerning interaction with the son of a stepmother born from adultery? Is he a forbidden (mahram) relative to the husband's children? Does he inherit? And how should one interact with this stepmother?

The aforementioned child is legally attributed to your father and can only be disowned through li'an (oath of condemnation). You and your sisters would be haram (unmarriageable) to him, unless your grandmother breastfed him, in which case he would become your paternal uncle through rada'ah (fosterage). The proof for this is the hadith of Sa'd ibn Abi Waqqas and 'Abd ibn Zam'ah, in which the Prophet, peace and blessings be upon him, ruled that "the child belongs to the marital bed," and he instructed Sawdah bint Zam'ah to observe hijab as a precaution. This child inherits from your father and his siblings, and your father's wife is haram to all her sons. The wife should be treated with good Islamic manners, and it is advisable to avoid meeting with her if possible.