Does the law of Universal Gravitation apply to every matter? If I have a big ball of 20,000,000kg and another of 100g, does it mean that the big ball will pull the small ball towards it?
 A: The magnitude of the force for both balls is $F = G \frac{M m}{r^2}$. Here $M$ is the mass of the big and $m$ the mass of the small ball, both assumed to be point masses. As you can see the force is the same on both balls.
So $F_{Big} = F_{small}$. This leads with $F = m\cdot a $ to $m a_{small} = M a_{big}$ or in a much nicer way:
$$\frac{a_{small}}{a_{big}} = \frac M m = 200,000,000$$
As you can see both balls will move, but the smaller ball has a much bigger acceleartion. With this you can assume that the big ball stays at rest and only the small ball is moving.
A: The gravitational force between the 2 balls is $F=\frac{Gm_1m_2}{r^2}=\frac{6.6\times 10^{-11}\times 2\times 10^7\times 0.1}{r^2}=\frac{1.32\times 10^{-4}}{r^2}$
So in order that the force be of 1 N, the distance between the two masses should be about $r=1~cm$. If the density of the larger ball is $\rho=10^4~kg/m^3$, its volume is $V=2000~m^3$ which simply means that its radius is much greater than $1~cm$ and that the gravitational force in this case is much less than $1~N$ and doesn't cause any noticeable effect.
A: Both balls would exert equal and opposite attractive forces on one another. But, the motion depends on environment (other forces may affect it).
Providing there's no other force, it may look like massive ball is pulling lighter ball. Its because massive ball would offer higher resistance to other's force. This causes massive ball to accelerate slower than lighter ball.
A: In a certain sense, you are correct, "the big ball will pull the small ball towards it".
Essentially, both the big ball and the small ball will accelerate towards the center of mass of the system which, in this case, is very close to the center of the big ball.
So, from the perspective of the center of mass, the big ball barely moves while the small ball rapidly accelerates towards the big ball.
