The iddah has ended, but my ex-husband suggests we keep living in the same house for the children's sake. Is that allowed?

Not in the usual way, because after the iddah the two of you are not mahram, and living in the same house opens the way to khalwat (being alone together) and slander. His intention may be good, but children need parents who are calm and whose relationship status is clear. Consider alternatives: living close by so the children can go back and forth easily, or a frequent visiting schedule. If finances force you to live in the same building for a while, the units must be truly separate and there must be no khalwat. Discuss the actual arrangement with the district Islamic religious office for certainty. If the two of you actually want to get back together, the proper way is a new marriage contract (akad nikah). See also mahram.