The first word "he" refers to Paul. The second word "he" refers to God. Here the word "our" refers to Paul and his fellow Jews. The words "they" and "them" refer to the Israelites. (See: [[rc://en/ta/man/translate/figs-inclusive]])
This could refer to moving his hands as a signal that he was ready to speak. Alternate translation: "moved his hands to show that he was about to speak" (See: [[rc://en/ta/man/translate/translate-symaction]])