1) При каком правителе Иван Сырков построил церковь Жен-Миронос

Based on the text, Ivan Syrkov built the Church of the Nativity of the Virgin in 1510. The text states that this was after the Novgorod lands were joined to Moscow. While Ivan III was the ruler who oversaw the incorporation of Novgorod into Moscow, the text does not explicitly state that he was the ruler at the *exact* moment Syrkov built this specific church. However, given the context of Novgorod's incorporation and the mention of Ivan III, it is highly probable that Ivan III was the reigning prince during this period. The text does not mention any other specific ruler in relation to Syrkov's early construction in 1510. Therefore, the most logical answer based on the provided information is Ivan III.