Clarification of the proof involving the regularity condition in Master Theorem
Let's start with the issue of iteration. Suppose that a function $f$ satisfies
$$ f(n/b) \leq (c/a)f(n). $$
Then it also satisfies
$$ f(n/b^2) \leq (c/a)f(n/b) \leq (c/a)^2 f(n). $$
You can prove by induction that for all integer $t \geq 0$,
$$ f(n/b^t) \leq (c/a)^t f(n). $$
As for your second question, about assuming that $n$ is large enough: the proof is just sloppy. You cannot assume that $f(n/b) \leq (c/a) f(n)$ holds for all $n \geq b$. Indeed, in Introduction to Algorithms, third edition, they do not make such an assumption for the case where $n$ is a power of $b$.
They do seem to make such as assumption in the case of general $n$, but what they are really saying is that the inequality $f(\lceil n/b \rceil) \leq (c/a) f(n)$ only makes sense for $n \ge b + b/(b-1)$. Using the idea of the proof of the special case where $n$ is a power of $b$, you can complete the proof of the general case. I would, however, strongly suggest ignoring such technicalities at present. The master theorem is essentially a calculation, and you can trust the authors that it works out. Nothing interesting is hidden under the rug.