I think this is supposed to be an example of one ray (that contains all points on the right side of the horizontal axis and the origin) which cannot have a unique pair (𝑟,𝜃) assigned to any of its points.
Yes, I think this is what he means. I was confused by this paragraph as well.
He doesn't really say it, but of course our choice of $\theta$ is somewhat arbitrary, so long as it covers the full 360°-$2\pi$ radians.
We can use $\theta$'s defined by
$$0 \leq \theta \leq 360$$
or
$$73 \leq \theta \leq 433$$
or
$$-\frac{3\pi}{4} \leq \theta \leq \frac{5\pi}{4}$$
(Analogously, we could choose any ray through the origin against which we measure $\theta$. The choice to use the positive $x$-axis is somewhat arbitrary.)
In general we can use
$$\theta_0 \leq \theta \leq\theta_0 + 360$$
where $\theta_0$ is any initial angle.
I think what he's getting at with his "exclude some ray through the origin"
is that we exclude all points along the ray defined by $(r,\theta_0)$ where $r \geq 0$.
For any choice of $\theta_0$, we exclude the points $(r,\theta_0)$, along with the origin, leaving us only with points that can be unambiguously assigned to unique $(r, \theta)$, where $\theta_0 < \theta < \theta_0 +360$ and $r > 0$.
Analogously, We can chose to measure our $\theta$ against the positive $y$-axis, or the ray that's 76° counterclockwise from the negative $x$-axis, or any other ray through the origin, but regardless of which one we chose we must exclude points along this ray including the origin if we want to be able to assign unique $(r, \theta)$ coordinates.
This of course is not really satisfactory and will leave us with points we can't describe regardless of our choice of $\theta_0$
We could instead, restrict our $\theta$ to either
$$\theta_0 < \theta \leq \theta_0 + 360$$
or
$$\theta_0 \leq \theta < \theta_0 + 360$$
and define the origin point as having some fixed $\theta$, for example $\theta_0$ or $\theta_0 + 360$, depending on our choice of allowed $\theta$. We can easily avoid the ambiguity Spivak's worried about, without excluding any rays through the origin.
Fortunately, it's not really important. He goes on to say that, despite any difficulties in picking points on the plane and then assigning to them unique $(r, \theta)$ coordinates, going the other way is no problem at all.
Given any $(r, \theta)$, this maps to a unique $(x,y)$ in the plane, and the rest of the chapter involves this process.
Question 1
Spivak simply multiplies the inequality by $-1$. When you do that, the sign on the inequality flips (you can look at Joe's excellent comment to see why). If you'd like, you can consider the compound inequality in parts:
$$x<y \implies -x>-y$$
$$y \leq 0 \implies -y \geq 0$$
Question 2
I'm not sure what you mean. The claim states that $n$ is odd; that's why Spivak mentions it. If you're wondering why $n$ has to be odd, it's because if you have $x<y$ but $|x|>|y|$ (for example, $-4<3$), an even $n$ will go against the claim ($16>9$, $256>81$, etc.).
Final Question
He isn't restating it: he first has $(-y)^n<(-x)^n$. Then, because $n$ is odd, we can write this as $-y^n<-x^n$ (i.e. odd powers of $-1$ are $-1$).
And yes; see my response to question 1. If $-y^n<-x^n$ then $x^n<y^n$.
Best Answer
Since Spivak only defines the meaning of $a^x$ (when $a>0$ and $x\in\Bbb R$) four chapters later, and since he uses the letter $r$, I think that you can safely assume that $r\in\Bbb Q$.