Under what circumstances is a divorced couple with children allowed to live in the same neighborhood?
If the neighborhood is a large, ordinary-sized neighborhood, it is permitted for the divorced couple to live there. The prohibition is only on living in the same yard (chatzer), or the same dead-end road. It is, however, customary to avoid the same neighborhood, though this is not a halachic obligation. [If the wife has remarried, the halachah is somewhat more stringent; see Aruch Hashulchan, Even Ha’ezer 119:26). In addition, some are more lenient in special circumstances, such as for the sake of bringing up the children (where necessary).
Sources: See Shulchan Aruch, Even Ha’ezer 119:7; Aruch Hashulchan 119:27; Get Pashut 119:46 (in the name of Ra’anach); concerning leniency with regard to children, see Be’ahalah Shel Torah, vol, 1, p. 408.