Every instance that you obey any law, you are keeping the commandments. It's really not about perfection. It's about direction. Hey that rhymes!
Proverbs 24:16 For a righteous man falls seven times, and rises again
Do you fathom the significance of that verse? It says two things. First, that the righteous are NOT perfect, that they fall, and more than once. But what distinguishes them from the wicked is that they "rise again." IOW repent.
So there are PLENTY of Jews that are righteous.
Not that I'm aware of. In my experience, with very few exceptions, Christians make no effort to obey the Law. They teach that Jesus "fulfilled the law" and interpret that to mean he abrogated the law (despite Jesus saying the exact opposite. They eat bacon and mow their lawn on Saturday and have no idea when Yom Kippur is.
Actually, you had plenty of refugees from the Northern Kingdom who fled south to Judah. There is plenty of archaeological evidence that proves this in fact. So basically, when the Kingdom of Judah went into the Babylonian Captivity, it included people from ALLLLLL the tribes, and in Babylon they were ALLLLLLL called Jews. Thus Hebrew=Israelite=Jew