Is it permissible, from a Sharia perspective, to take back a divorced wife in order to preserve the family unit and raise the children, without cohabitation or spending the night together? And is this considered cuckoldry?

If it becomes apparent that the woman has repented sincerely, there is no impediment to returning to her; for repentance erases what came before it. If she waives her rights to maintenance (nafaqah) and overnight stay (mabeet), there is no blame upon you, but she is permitted to retract what she has waived. It is preferable, if you take her back, that you fulfill her right to conjugal relations to keep her chaste. However, if it becomes apparent that she has not repented or has returned to unlawful acts, you are not to keep her, and divorce may be obligatory in this case.