Regarding the natural place to introduce e, my preference is to wait until calculus. Once you learn that the antiderivative of x^n is x^[n+1]/(n+1), it becomes fascinating to ask: what happens when n = -1? So define L(x) as the indefinite integral of 1/x and explore it.

Doing it this way, you discover something truly amazing and beautiful: L(x) behaves like a logarithm! For instance, it obeys L(ax)=L(a)+L(x), as you can show by taking d/dx of L(ax). Once you know L is a log function, the natural question is: what is its base? Answer: call it e

Then, once e is in hand, define e^x as the inverse function to L(x). After that, you discover further wonders: e^x is its own derivative! Or as my old HS calc book (by Lynch and Ostberg) put it, e^x is "indestructible" under differentiation.

And only ofter that do you discover the cool identity that e = lim (1+1/n)^n as n tends to infinity. At that point, you're ready to do continuously compounded interest problems.

Maybe not to everyone's taste, but this is how I like to teach e and related concepts. Doing it this way, it all feels like a wondrous voyage of discovery -- which it is! -- and much more motivated than the current fashion of shoehorning e into alg 2 or pre-calc.

Incidentally, this is closer to how ln(x) and e were discovered historically. They arose in the challenge of finding the "quadrature of the hyperbola" (i.e., finding the area under y = 1/(1+x), in modern language), solved by Newton and Mercator. See INFINITE POWERS for discussion