A couple is married in the Church, get a civil divorce, don’t get an annulment, live chastely after the divorce, and then reconcile. Can they have intercourse since they are still married in the eyes of the Church?

For a Catholic, divorce is a purely civil matter and does not affect the validity of a sacramental marriage whatsoever. The husband and wife will always retain the right to express their marriage vows sexually. When such individuals reconcile, they are certainly free to live as man and wife.