How can a worshiper ensure the correctness of the imam's recitation, especially Al-Fatiha, and his proficiency in prayer generally, when entering a mosque for the first time, and what should he do if he is not reassured by that?

The default is the validity of Muslims' prayers. Therefore, pray behind whoever you find praying, unless you know or strongly believe that he is violating a condition or a pillar, in which case it is not permissible to follow him, and your prayer alone is preferable.