I agree with your understanding on everything other than the New Covenant. Actually it was the Old Covenant RENEWED. Same Ten Commandments, same Covenant, only renewed. Look up New in Strongs...renewed in the Greek.
That's correct. The Covenant is with the Jewish people and Jewish people only, and cannot be abrogated or annulled, replaced, or "made obsolete," as the Christian book of Hebrews (Ch. 8) suggests. The Covenant God made with Israel is an everlasting covenant; the Torah is to be observed forever, and therefore cannot be "fulfilled."
God also made an everlasting Covenant with the Gentiles after the Flood. Therefore a "new covenant" proposed by Christians is in conflict with God's everlasting covenant made to Noah. The Gentiles have been given the Noahide laws and that's where their obligation and dealings with the God of Israel ends.