I know the Torah claims the Jews descend from a single patriarch Abraham (b. ~1800 B.C.E. according to the literal reading of the Torah), and formed a nation upon the Exodus from Egypt and the Revelation at Sinai (~1300 B.C.E.). According to scientific evidence- archaelogical, documentary, or other (maybe even contemporary verifiable accounts from other sources)- what is the earliest time and place that we definitely know they existed? And in what form- as a nation, religion, ethnicity or other.
The first verifiable mention of Israel comes from the Merneptah Stele in Egypt, recounting Merneptah's victories over the lands around him. One of the lines of the stele is "Israel is laid waste and its seed is no more." This stele dates to roughly 1208 BCE. Not exactly what you're looking for, but the closest answer I am aware of.
Funny you should ask this as I just posted something about this a few days ago. Below is a snippet by world renown Egyptologist Eric Cline who wrote about the earliest mentions of Judaism outside of the Torah in one of his books, saying:
Merneptah is perhaps best known to students of the ancient Near East as the Egyptian pharaoh who first uses the term “Israel,” in an inscription dating to this same year (1207 BC). This inscription is the earliest occurrence of the name Israel outside the Bible. In the Pharaonic inscription, the name—written with a special sign to indicate that it is a people rather than just a place— appears in a brief description of a campaign to the region of Canaan, where the people whom he calls “Israel” were located.
Source: Cline, Eric H. (2014-03-23). 1177 B.C.: The Year Civilization Collapsed (Turning Points in Ancient History) Princeton University Press. (pp. 6-7)
I'm a grad student studying early Christian history but I also have studied other areas around ancient Mesopotamia and Near East and have studied ancient Israel as well so I am a bit well versed on the matter. If you have any further questions please let me know.
I actually just recently watched this PBS documentary about the origins of the Israelite peoples. It's really quite fascinating, and I think will help to answer some of the questions you might have, albeit over the course of 100 minutes or so.
But to answer your question, the basic answer is that, as /u/PrincessAnika pointed out, the Merneptah Stele is the earliest known mention of the existence of a group of people known as Israel.
You're using Jew and Israelite synonynously, when they never were so in the Bible. Jew refers to the tribe of Judah, which is only one of the twelve tribes. I'm not even sure if Jew in the religious sense (rather than Judahite) is an appropriate term before the Babylonian exile of thwle sixth century, as Jew implies common membership in a faith community that many even in the Judahite kingdom (King Manasseh and King Amon) manifestly were not part of. Judaism as a religion isn't really in existence until the very end of the Old Testament era.