How is the radian measure of angles derived/defined?

(2):

Since $C=2\pi r$, if you double $r$, you double $C$ and you also double "a third of $C$", etc. This works with not just doubling, but whatever scaling you need to get from one circle to another. $C/r$ is always $2\pi$, for any circle, so $(1/3)C/r$ is always $2\pi/3$, etc. Therefore, if you have an arc of length $s$ that's a certain ratio of the whole circumference (like 1/3), then $s/r$ is the same for all circles (in this case it's $2\pi/3$).

(1):

An angle from the center of the circle always covers a ratio of the whole circle that doesn't change, so if you care about angles, you care about fractions of the circumference. To see this, it might help to look at a picture like:

(Picture taken from http://ck022.k12.sd.us/)