Am I permitted to marry my second wife, legally and officially, after divorcing her three times in an unregistered marriage? And is unregistered marriage forbidden and unrecognized?

If the 'Urfi marriage (customary marriage) is religiously valid (with an offer and acceptance, mahr, and witnesses) but is not officially documented, then a divorce in such a marriage is effective. In this case, you are not permitted to remarry the woman after divoring her three times until she marries another man, with the prohibition of tahlil marriage (a marriage intended to make a divorced woman lawful for her previous husband).

However, if the 'Urfi marriage was concluded without the guardian's permission, scholars have differed regarding its validity. According to Abu Hanifa, the marriage is valid. But the majority of jurists consider it invalid based on the Prophet's (peace be upon him) saying: "Any woman who marries without the permission of her guardian, her marriage is void."

Based on both opinions, if you believe in the validity of the marriage, then the rulings of a valid marriage apply to you, and your divorce is effective and counted against you. You are not permitted to trifle with Allah's rulings by seeking to invalidate the marriage in order to nullify the divorces. Therefore, it is not lawful for you to marry this woman until she marries another man.