Is taking a separate apartment for the wife considered disobedience to the mother, especially since the mother stipulated to the husband that she share the home, and the spouses were unable to live together peacefully?

A wife has the right to independent housing. This is not disobedience to the mother, but rather giving every rightful person their due. The husband must not neglect his mother and should be gentle with her. It is preferable for his residence to be close to hers so he can care for her. Providing a servant for the mother, if possible, is considered an act of righteousness towards her, for her contentment is a gateway to Paradise.