Is divorce before an infidel judge considered a divorce if there was no intention of divorce? And is it permissible to have intercourse with a woman who has been revocably divorced without the intention of taking her back?

If a man signs a divorce document with intent, his divorce is effective. If the divorce is revocable and the husband affirms it before a non-Muslim judge by saying "yes" or similar, for the purpose of administrative disengagement, no new divorce occurs unless he utters an explicit divorce pronouncement or an ambiguous phrase with the intention of divorce, and his statement would then be a lie. This ruling is not exclusive to a non-Muslim judge. It is permissible for a husband to have intercourse with his revocably divorced wife, and her return is valid by that according to some scholars, but it is preferable that intercourse be with the intention of reconciliation to avoid scholarly اختلاف.