Is the conversion of a Christian to Islam for the purpose of marrying a Muslim woman considered a mistake? And is his conversion to Islam accepted if it is for the purpose of distancing himself from his family to avoid social problems, and to continue his life in another country?

It is by Allah's grace that a person's heart is opened to Islam. Perhaps your desire to marry a Muslim woman is a reason for you to embrace Islam, and something similar happened to some of the Companions. This is seen in the story of Abu Talha's conversion to Islam when he proposed to Umm Sulaym, and she stipulated Islam as her dowry. Also, Ikrima ibn Abi Jahl, whose wife Umm Hakim embraced Islam before him and sought protection for him. Many entered Islam for worldly purposes and then became among the most loving and adherent people to it, having witnessed the purity of its creed. The lesson lies in the initial intention; if the intention is for Allah Almighty, then whatever accompanies it does no harm.