Someone believes in Judiasm, lives by all the rules, observances etc, then do you think God would be bothered about whether or not they'd passed an exam/assessment etc to officially become Jewish?
No, one is born a Jew, so if your mother isn't Jewish, then you're not Jewish. Religious observance has no bearing. I know some Jews who don't consider those whom have converted to be Jewish, even though they (those whom have converted that is) tend to be incredibly observant and religious.
If Jewishness is passed on through the female line, what would happen if someone discovered, for example, that their grandmother technically wasn't Jewish? Does that mean her children, and grandchildren aren't Jewish either? If so, how many generations could this go back?
What do you mean someone isn't technically Jewish? If your mother isn't Jewish, you're not Jewish. So if my mother's grandmother wasn't Jewish, then she wouldn't be Jewish, nor would I or my children. However, if my father's mother wasn't Jewish, then it doesn't matter so much.