Is it permissible to pray in a mosque where the regular imam does not lead the prayers, leaving anyone to lead us, with some muezzins having Tajweed and grammatical errors, and is the prayer valid in it given the presence of other mosques in the area?

It is permissible to pray in the aforementioned mosque if the regular Imam grants permission to someone else to lead the prayers. Praying behind someone who makes mistakes in Tajweed rulings is clarified in Fatwa No. 145243. If the grammatical error (I'rab) in Al-Fatihah alters the meaning, then it is not permissible to follow him in prayer. However, if the error is outside Al-Fatihah, or if it does not alter the meaning within it, then the prayer is valid.