they didn't "become" Hebrew. They started out as Hebrew, because Abraham was a Hebrew.
Hebrew people later became called Israelites, then Judaeans, then Jews. Today they are called Jews.
According to the Hebrew Bible, he lead the Israelites out of Egypt.
The Israelites spoke Hebrew, as is abundantly clear from the Hebrew Bible.
there's no such designation as a "back Hebrew Israelites"
for the most part
Hebrew, until the late 1st Millennium BCE, when Aramaic (introduced by Persia) and and Greek.(by Alexander) became common. Hebrew became a ceremonial language used in daily prayer and Torah-learning, continually throughout the centuries.
It was written in Hebrew
No, Joshua became the leader of the Israelites after Moses died.
The Israelites, who were lead by Joshua.
Hebrew Israelites
Yes, Hebrew on parchment. The Israelites always had a high level of literacy. The Ancient Israelites used a derivative of the Phoenician alphabet to transcribe Hebrew. Eventually, the Hebrew derivative of the Phoenician alphabet became particularized to Hebrew and became the forerunner of the Modern Hebrew alphabet.
They were called Hebrews or Israelites.