To what extent is obedience to non-Muslim parents obligatory if their obedience conflicts with moving to a Muslim city, and what is the ruling of Islamic law regarding the lineage of a son born out of wedlock?

You must honor your disbelieving parents, treat them kindly, and accompany them in goodness, even if they are polytheists. However, you are not obligated to obey them if they order you to abandon moving to a place that benefits your religion. If their intention is to deter you from Islam or its obligations, then you must move to the place where there is benefit for your religion and for your husband, while continuing to invite them to Islam with wisdom and kindness. As for the child born from an unlawful relationship, he is not attributed to you but rather to his mother, unless the mother is unmarried, in which case you may claim him (i.e., attribute him to yourself).