Effect of Newton's law to the orbits inside a galaxy like Milky Way Does Newton's law describe the orbital motion of stars in a galaxy? Is it true that the closer a star is to the center, the lower is the gravity, so the lower the speed of the star?
 A: 
Does the Newton law describe the orbital motion of stars in a galaxy in that way that the closer a star to the COG is the lower the gravity so lower the speed of the star?

Short answer yes, Newton's law applies in galaxies.
Long answer, Newton's law alone does not imply that a closer star to the COG will have lower gravity. And lower gravity alone does not imply that the speed of the star will be lower.
The main difference between Newton's Law for stars in galaxies vs. Newton's law for planets in the Solar system is the distribution of matter.
For circular orbits and spherically symmetric matter distributions, you can use $v^{2} = \frac{GM(r)}{r}$ where $M(r)$ is the mass enclosed in radius $r$.
In the Solar system, $M(r)$ is roughly constant because most of the mass is in the Sun, which is at the very center. So planets with smaller orbits experience more gravity and have higher velocity.
In a galaxy, $M(r)$ depends on how the mass is distributed.
When the density is roughly constant, $M(r) \propto r^{3}$ so $v^{2} \propto \frac{r^{3}}{r}$ so $v \propto r$. In this situation, it's true that closer stars experience lower gravity and lower speed.
When the density is roughly $\rho \propto r^{-2}$ as in an NFW halo then $M \propto r$ so $v^{2} \propto \frac{r}{r} = 1$ for constant velocity. In this case, closer stars experience larger gravity $\propto \frac{M}{r^{2}} \propto \frac{1}{r}$ and the same speed.
Imagine $M(r) \propto r^{1.5}$. Then the gravity would be $\frac{M}{r^{2}} \propto 1/\sqrt{r}$, larger for closer stars. But $v^{2} \propto \frac{M}{r} \propto \frac{r^{1.5}}{r} \propto \sqrt{r}$. In this situation, the gravity is larger for closer stars, but the velocity is actually smaller.
This is because equating centripetal acceleration with gravity $\frac{v^{2}}{r} = \frac{GM}{r^{2}} = F$, $v^{2} = Fr$, so velocity and force of gravity do not always increase or decrease together.
A: Yes.
Orbits of stars in galaxies are classical and well-described by Newton's laws, taking into account the distribution of matter in the galaxy.
A: Yes, but only within the 'central bulge'.  According to Newton's gravity the speed of a star would follow the red curve below.

Newton's gravity is an inverse square law, but the amount of matter inside a sphere of radius $r$ depends on $r^3$, (if the density is constant). So for low radii, within the spherical 'central bulge', the speed is approximately proportional to radius
$$\frac{GM}{r^2} = \frac{kr^3}{r^2} = \frac{v^2}{r}$$
where $k$ is a constant, so
$$v^2=kr^2$$
and $v$ is proportional to $r$.
For higher radii, the speeds have been measured to be different to what Newton's law predicts, if there were only the visible matter - hence the 'dark matter' proposal.  With dark matter included, Newton's law gives the correct values for the velocity of stars at all radii.  However at these larger radii the velocity is approximately constant as the radius increases, (green curve).
