Egg vs. chicken: trig functions, exponential, real and complex

This is something I was shaky about when I took calculus, real analysis, and then complex analysis. Specifically, is the following chain of definitions circular in any way?

1. Define the set $\mathbb{N}$ of natural numbers, define basic arithmetic operations


2. Extend $\mathbb{N}$ to $\mathbb{Z}$, then to $\mathbb{Q}$, and extend the arithmetic operations to these sets as well


3. Use the arithmetic operations to define distance and use that to define convergence, limit, and related concepts


4. Use 3 to extend $\mathbb{Q}$ to $\mathbb{R}$



5. Define cosine and sine on $\mathbb{R}$ using the series definitions and then define other trig functions using cosine and sine


6. Define $e=\lim(1+1/n)^n$ and define $\pi$ as $\ldots$


7. Use 5 and 6 to derive the usual properties of trig functions, of $e$, and of $\pi$


8. Define $e^x$, $x$ real, as $\lim(1+x/n)^n$


9. Finally, define, for $z=x+iy$, $e^z=e^x(\cos y+i\sin y)$, and derive properties of $e^z$ from 7 and 8 and complex arithmetic operations.

I also have 3 auxiliary questions: (i) what is a standard / convenient definition for $\pi$ above? (ii) how does one fork off from 5 and 6 above to get the geometric interpretations of $\pi$ and the trig functions? (iii) is there a book (or a few books) that give students a sort of big picture ("big" for a guy like me) view above?