Is the previous unregistered marriage between the questioner and his wife considered void or valid, and does it affect the validity of their future legitimate marriage with the consent of her guardian, and do the three previous divorces in that unregistered marriage count?

Marriage without a guardian (wali) and with only one witness is an invalid marriage (nikah fasid). Divorce in an invalid marriage is binding if it is pronounced, and it results in an irrevocable (ba'in) divorce that does not allow for a return (raj'ah).

So, if you divorced this wife of yours for the first time, she became irrevocably separated from you with a minor irrevocability (baynunah sughra). She is not permissible for you except through a new contract. The two subsequent divorces pronounced after that are not taken into consideration. You may contract a new marriage with her, fulfilling all the conditions, and it will be considered as one divorce, leaving her with two remaining divorces.