About FM signal's proving
As you have correctly derived, the difference of carrier and instantaneous frequency is

$$
g(s(t)) = \omega(t) - \omega_\mathrm{carrier} = 2\pi k s(t) + 2\pi k s(t)'t
$$

I will omit $t$ for convenience. If $g(s)$ is a linear function of $s$ it must fulfill the following condition:

$$
g(c(a + b)) = cg(a) + cg(b)
$$
So here:
$$
g(c(a+b)) = 2\pi k c(a + b) + 2\pi k \left[\frac{d}{dt}(c(a + b))\right]t\\
= 2\pi k c a + 2\pi k c b + 2\pi k c a' t + 2\pi k c b' t\\
= c(2\pi k a + 2\pi k a' t) + c(2\pi k b + 2\pi k b' t)\\
= cg(a) + cg(b)
$$
Thus, $g(s(t))$ is indeed a linear function of $s(t)$.