I don't think you see the word "Jewish" in the Jewish Scriptures. The term comes from the an earlier term for Jews, "Judean" because after the Assyrian conquest of the Northern Kingdom, all remaining Israelites who weren't exiled lived in Judea - the Judeans.
There are three major Jewish denominations (in progressively more liberal order): Orthodox, Conservative and Reform. There are other smaller ones like Reconstructionist and Humanist.
From an historical perspective, Karaite Judaism is also a form of Judaism, although not often viewed as such. The Samaritans were also once believe to be Jews, but that later changed according to the Talmud.
Yes. The main difference is the approach to Halacha, Jewish Law. Orthodoxy believes it to be binding. Conservative believes it to be binding but also believe they have the power to remove restrictions or circumvent Laws based on modern philosophies. Reform do not believe Jewish Law is binding but a matter of personal choice.
Within Orthodoxy, there can also be differences in practice between ultra-Orthodox and Modern Orthodox movements or even within the various sub-sects of ultra-Orthodoxy. Those differences though are most often based on customs not Law, either from those that arose in different regions or because of added emphasis on different parts of Jewish Law.
Karaite Judaism is basically a Saducee Renewal sect. They reject the authenticity of the Oral Torah and favor their own approach to interpretation.
No. Identifying as Jewish for Orthodox and Conservative sects requires that one's mother be Jewish or that one converted. For Reform Judaism, it requires that either of one's parents be Jewish and that one was raised in a Jewish environment or that one was converted.
For Orthodoxy, all more liberal strains of Judaism (ie. all the other ones) do not fulfill the Halachic requirements in converting and likewise does not accept their converts as Jewish. Conservative will generally accept Orthodox converts but not necessarily accept Reform converts.
An Israeli is someone who has citizenship in the country of Israel. There are Israeli Christians, Israeli Muslims, Israeli Druze.
An Israelite is an older term for Jew.
No, the term Noahide is used today to refer to those who follow the 7 Noahide Laws as prescribed by Jewish Law for non-Jews. So it specifically refers to those who haven't converted to Judaism. A non-Jew who converts wouldn't be a Noahide, but a Jew.
Sometimes non-Jews believe themselves to be Noahides when in fact they've divorced the term from it's Jewish context and applied it according to their own reasoning. The source of the concept comes from the Talmud and it comes with it's own requirements
as expressed here. It's far more common to find a self-defined Noahide than one who actually fits the requirements.