Lets assume that the motions of the Moon around the Earth and of the Earth around the Sun are uniform circular. If $\omega_L, \omega_T$ are the respective angular speeds find $\omega_L/\omega_t$.

Since the motions are uniform circular we have $\omega = v/R$ with $R$ radius, hence $\omega_L/\omega_T=\frac{R_T}{R_L}\cdot \frac{v_L}{v_T}\simeq \frac{1.5}{4,0}\cdot 10^3 \cdot \frac{1,0}{3,0} = 125$.

If the planes of the motions are the same, is the angular speed of the Moon with respect to the Sun constant?

I think it is not. Let $r_1$ be the vector that connects the Sun and the Earth and $r_2$ be the vector that connects the Earth and the Moon, if $r_1 \perp r_2$ and we consider a rotation of an angle $\pi$ when $r_2$ is directed "as the motion of the earth" and when is "opposed to the motion of the earth", then the variation of the angle between the Sun and the Moon is bigger in the last case.

I've tried to prove my claim but I have some trouble. Is it correct so far? Is it sufficient my explanation? Otherwise can you give me a hint?

Note: sorry if I have not explained myself very well, English is not my native language.