If Eve is the mother of all the "living"
then it means her husband isn't alive or she gave birth to her husband on her own
this also brings to question the nature of Cain (whom was "acquired") and Abel who was born.
If you see it like I do (Jacob wrestling with the Angel) it is possible that the true origin is not where it appears.
Take it a step further...God said to Noahs sons that the fear of all beasts will be on them and that everything that moves would be food for them:
Shem (The Name), Ham (Black) and Japheth (May he expand. Enlargement. May He grant ample room)
Reading it the way I read it....Noah (Rest) OR Shem is Eve
Adam represented everything before G-d split Eve from him, right? I don't think it would necessarily contradict your hypothesis of Eve giving birth to all life but not Adam.
Eve eating of the apple, to me, was both her willingness to adhere to the perception/reality of the serpent moreso than to G-d. A good analogy is a huge field; you may see a garden, but I see a nation. Two separate realities for one space.
The fall of man is giving into the reality outside G-d because, while that reality is true, it will always be a step below.
-o-
As for the true origin, from our vantage point there seems to be a strict starting point, but just as in learning about relativity, time depends on your frame of reference.
That frame of reference changes based on how you measure (level of consciousness). So there may be some legitimacy to what you said. I still have to learn Hebrew though.
-o-
Something interesting that you touched up on that I have been thinking about but can't really put into words:
Shem (the name) - Speech
Ham (Black) - darkness, reality, possibility; it is where G-d exists:
"And the earth was without form and void, and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters." Genesis 1:2
"God is surrounded by clouds and thick darkness, yet His righteousness and justice are the foundation of His throne." - Psalm 97:2
Japheth (enlargement) - space or "life" depending on the interpretation
I remember a pastor saying we should read the Bible as a tool to understand how the spiritual realm works and the rules of that realm. Every story details a rule/lesson of how the spirit works.
Your post kind of confirms that to me.