Does a non-Muslim brother's failure to pray render him unqualified to be a guardian (wali) for his Muslim sister in marriage? What is the procedure if there is no Muslim guardian from among the relatives, and can the imam of the mosque act as a judge in marrying off a Muslim woman residing in a non-Islamic country? And does the lack of spousal compatibility affect the validity of the contract?

If your brother denies the obligation of prayer, then he is a disbeliever (kafir) and it is not permissible for him to be your guardian (wali). If he acknowledges its obligation but abandons it due to laziness: if you follow the opinion of those who deem anyone who abandons prayer a disbeliever absolutely, then it is not permissible for him to be your guardian. However, if you follow the opinion of those who do not deem him a disbeliever, then there is no impediment to his guardianship.

If none of your asabat (male relatives on your father's side) are suitable for guardianship, then the one who marries you off is the Muslim Sharia (definition) judge. It is not permissible for the Imam of the mosque or anyone else to marry you off as long as there is a judge in the land. If a woman has no guardian and no one in authority, then a just man may marry her off with her permission, provided he takes precautions regarding suitability (kafa'ah) and the dowry, especially if there is no judge in the area.

The condition of kafa'ah (suitability/compatibility) for the validity of marriage is a matter of dispute among scholars, and most of them hold that marriage without kafa'ah is valid.