It's all vulgar speculation, so I'll add to it.
I doubt the pills have anything to do with mental stability or control (again, if that's good for Rei, it would be used on the other pilots). It's more likely the pills are for physical maintenance, and seems to be one of the few explanations for her arm falling off (she's physically unstable).
There seems to be a lot of identity placed on soul per se here. If we stipulate some of the stylistic influences of NGE to be somewhat more meaningful than mere window dressing, then the soul is not identity ab initio. It is just the motivating force, or "breath", of life - literally, that energy that causes breathing. Eastern thought, and Western mysticism (as opposed to more garden variety thought), point toward development of the soul through life experiences.
In other words, Rei without the experiences she has as a human is not Rei as we know her. Her identity is a result of those experiences. Without those, she is not Rei any longer. She can never be a normal human, because she has experiences as a SOL. It is experience (not memory as such) that is stored in the soul in this view.
The soul is not "who you are" from beginning to end; the soul is molded by "what you do".
Rei, in an alternate universe, is not Rei. Rei with a happy home life is not Rei. Ve becomes just a paper doll stand-in for "Gee, wouldn't it be nice if all little girls had proper mommies and daddies," which is a nice thought but has nothing to do with Rei. Take any blank slate and put them in a happy home and see what happens - but it's not recongnizably Rei any more.
On her age, well, her entire human life could be only the few months we actually see her. There's no evidence she existed outside of her Reiquarium from 0 to about 5 years old, and then from being strangled to about 14 years old.
As far as babies are concerned, my recent experiences IRL suggest many people are still babies well into their 20's.
"I would like to see a clown remake of 'Terms of Endearment' or 'The Thorn Birds.' Or maybe a big disaster movie, like 'Towering Inferno.' That's stuff I'd pay to see. Nothing says entertainment more than burning clowns."