Can a man marry his Christian ex-wife whom he divorced three times with an 'Urfi divorce, and who desires to embrace Islam and return to him?

If the 'urfi' marriage (customary marriage) was valid, then the three divorces are considered to have taken place, and the wife is not lawful for you until she marries another husband. Her conversion to Islam does not change the ruling regarding the divorces. However, if the marriage was invalid and void, then the divorces do not occur, and repentance from zina (fornication/adultery) is obligatory. You can marry her after her conversion to Islam and her good conduct, with a guardian (wali). If there is no Muslim guardian, the Sharia judge can marry her, or she can appoint a trustworthy man as her representative. If she remains a Christian, your marriage to her is permissible if she is chaste, but it is preferable to avoid this and seek a righteous Muslim woman. As for marriages whose invalidity is debated (such as marriage without a guardian), their effects are still incurred.