Does the most recent divorce count if it was made in anger and without resolve, and is it considered the first or third divorce, in a marriage that took place after the wife embraced Islam and the marriage was documented, following two previous divorces, one of which was in extreme anger and the other intentional?

Marriage without a guardian (wali) or witnesses is invalid. What precedes a valid marriage contract is considered explicit fornication (zina) and requires repentance. The previous divorce is void due to the marriage not having been contracted in the first place. The contract after her conversion to Islam will be valid if its pillars are met (guardian, witnesses, صيغة - proper form/pronouncement). An explicit divorce pronouncement takes effect even if you did not intend it. A nuanced (kinaya) pronouncement takes effect if you intended it. If you were not aware of what you were saying during extreme anger, then the divorce does not take effect. A divorce that occurs after a valid marriage contract is considered a first divorce (talaq (definition)).