Possible meanings are 1) they were speaking the proper name of the God of Israel or 2) they believed that Israel worshiped one of many gods, "the god of Israel." See how you translated this in [1 Samuel 5:7](../05/07.md).
The words "by all means" are a forceful way of saying something. Alternate translation: "you must send a guilt offering" (See: [[rc://en/ta/man/jit/figs-idiom]])
Here "hand" is a metonym used to represent God's power to afflict or discipline. Alternate translation: "why he has not relieved your suffering" (See: [[rc://en/ta/man/jit/figs-metonymy]])