Yes I know if your Mom was Jewish you were, but ever wonder why?? I do since its the men who were allowed in the synagogues, not women. I think its because to grow the community and the men mated with more than one woman so every baby born was a Israelite who practiced Judaism , according to your OT, the Israelite men liked and mated and had children with non Israelite women. My birth cert says Caucasian American, now you might be an Israelite as you were born in Israel and you might be a jew if you practice Judaism (some form of it).
Who told you women are not allowrd into syangogues? You're misinformed.
I'm not sure I understand the logic you're talking about, but the logic behind it is that if a father of the child is unknown, or unsure, then the child's legacy is uncertain, too. In case it's the mother's genetics that counts, there is no doubt.
To be a Jew, you need to be born to a Jewish mother, or convert. Those are the conditions. Of course, that should be your practice, and not
Avoda Zarah. As for myself, since my birthmother wasn't Jewish, and my adoptive parents converted me, for converts children the law states differently. You are raised a Jew, and when you reach maturity you can choose whether you accept the Jewish religion/nationality to be part of you, or you want out. Natural born Jews don't get to choose, converts children do. I think you realize what my choice was.