Is it valid for a husband to take back his wife without her knowledge, and how can the wife ascertain that the reconciliation has occurred, and is she still considered to be under his marital authority while she is in her family's home, and if the husband officially divorces her now, would this be considered the third divorce?

If this divorce is the second, your husband may take you back during the waiting period (iddah) without requiring your consent or knowledge. It is recommended for him to inform you of it and to have witnesses to it. If the husband claims that he took you back during the waiting period, his statement is considered valid. The documentation of a divorce is not considered a new divorce unless the husband pronounces another divorce. In case of dispute, the Sharia judge will rule on the matter.