That is correct. The Hittite civilization proper collapsed around 1200 BC, about the time Israel were conquering Canaan. The Hittites mentioned in the Bible (like Uriah the Hittite) were most likely individual descendants of the Hittite empire. I remember reading recently that in the First Temple period, there was at least one village in Israel (in the Samaria region) that was populated by Hittites.
Nope. The OT Hittites are just a name, and it was borrowed for a then-newly discovered civ. And the conventional chronology is wrong.