Is a marriage that took place after the husband told his wife, "If you do this, then you are divorced," and the wife did it, then the husband claimed he was only threatening, considered a valid marriage, or is it a forbidden marriage? And what are the implications for the children of this marriage?

What your husband uttered is a conditional divorce, contingent upon a specific action, which you have performed. This constitutes a broken oath (tahneeth) on your husband's part. The majority of scholars hold that the divorce takes effect, whether the husband intended divorce or merely intended to threaten.

However, if the husband intended merely to threaten, and acts upon the opinion of those who rule that divorce does not occur in such a case, following that opinion by way of imitation (taqleed) and not out of personal desire, then there is no blame upon him in that. In such a scenario, the marital bond remains valid, cohabitation is permissible, and the children are attributed to both of you.