Do prayers that were offered facing the wrong direction for a year have to be repeated due to an error in applying the direction-finding method, and is it obligatory to perform all the prescribed prayers in the mosque despite its distance from the place of study?

If a worshipper strives to determine the Qibla and prays, their prayer is valid, and they are not required to repeat it, even if it later becomes clear that they were mistaken. However, if they pray without making an effort to determine the Qibla, or if they were able to inquire about it but did not, then they must repeat the prayer. A slight deviation from the Qibla does not invalidate the prayer.