Does the second and third divorce take effect if the husband believes them to be obligatory, and is his wife, with whom he has children but has not seen for two years, permissible for him?

If you did not pronounce the second and third divorces except based on the belief that it was obligatory due to news that turned out to be false, then the predominant view is that this divorce does not occur. This is because if someone bases their divorce on a reason that later proves to be nonexistent, their divorce does not take place. If this is the case, then the divorce is null and void, and your wife is lawful for you with the first marriage contract. The matter does not require a reconciliation or a new contract. However, if you did not take your wife back during her waiting period (iddah) from the first divorce (which is three menstrual cycles), then your wife would have become irrevocably divorced from you with a minor irrevocable divorce (baynunah sughra), and she can only return to you with a new contract fulfilling the conditions.