If Eve never sinned, and humans were indeed to be immortal, then how would this planet sustain the over population that would have occurred over the last thousands of years? If they were really asexual before sin, then there would have never been any lineage. No Cain, Abel, Seth, Noah, or Abraham, and essentially no story to tell in the Bible. There are different interpretations of pre-Fall and post-Fall sexuality according to Genesis, but it boils down to the need of sin and the Fall for the story of the Bible to come to fruition.
Some say that there was no real sexuality before the Fall, and that the very concept of marriage, husband/wife, family, having kids, etc. comes from the first sin. If that were the case, then there would indeed be no Bible at all. Others say that there was sexuality pre-Fall, and that the "shamefulness" that Adam and Eve felt regarding their bodies was actually shame before God, not each other. However, if this were the case, then the planet would ultimately overpopulate to the point of catastrophe, and the resources and wildlife would have suffered much earlier in the timeline. Either way, it seems like an interesting catch 22. Maybe God would have made more space and resources for us as saw fit, seeing as how it was Paradise and we didn't sin. But then comes the idea that eventually, SOMEONE would have sinned. Some kid, probably Adam/Eve's first, would have done something that all kids do, and it'd be considered sinful.
After the Fall, God essentially has the first legal trial, via Adam and Eve vs. God, and He sets forth some rules/conditions -
(a) I will greatly multiply your pain [labor] in childbearing;
(b) in pain [labor] you shall bring forth children,
(c) yet your desire shall be for your husband,
(d) and he shall rule over you.
The B is what makes it seem like there would be no childbearing before hand, and that this whole husband/wife concept is a result of the sin.
Anyways, it's all very interesting stuff. But to think it all happened because someone ate a piece of fruit, like *that's* the first original sin - not murder, theft, or anything like that, but eating fruit when told not to. That's the very essence of free will in itself, but it damned all mankind to death and painful births of kids.