Do we apply force to an object we are touching? Since every object that has a mass induces gravity, do we actually apply some very very small force on an object (for instance a feather) when we put our hands close to it, eventually touching it, a force to attract it towards us (since we are bigger)?
 A: Yes, you apply a force on a feather when you go near it. In turn, feather applies equal force, but opposite, on you. This is not true because of Newton's 3rd law but by the construction of gravity theory by Newton. He could have defined gravitational force as follows, $$F=\frac{GM^2}{r^2},$$
but soon realize that nature does not follow this law. The fact that $$F=\frac{GMm}{r^2},$$ has dire consequences. It implies that acceleration of a body in gravity is independent of its mass. It's not a trivial statement. In fact, this fact was one of the inspiring thought for Einstein towards General Relativity. As for the direction, if one body is attracting 2nd body, the 2nd body also has to attract 1st body by symmetry.
Now one would naively believe that feather will come towards you instead you accelerating towards feather. This is true, but not because we are exerting bigger force. We simply have more inertia against the motion as compared to feather. The acceleration of feather is $F/m_{feather}$. Since mass of feather is much less than our mass, it will accelerate much faster than us.
However, in real life that's not the case. The reason is that even though the mass of the feather is really low, the force acting on it is not sufficiently high. The gravitational force on feather as to counter friction, air resistance, gravity by earth, and many other forces. Gravity of our hand is not sufficient to go against these forces.
A: well, every body is experiencing force whether it is gravitational or other type of force. When we touch a body, the body may or may not move so we can eliminate work done but we cannot eliminate force, because the force might be very weak and may not change the shape/size of the object/body but the body experiences the force.
