Objects falling at the same speed ?

The gravitational force causes acceleration and the inertial resistance retards acceleration. So, the mass cancels out.

acceleration a = weight / inertial mass = m'*g/m" 

where :

m' is the gravitational mass

m'' is the inertial mass 

since m' and m'' have been found to be equal , the acceleration a is equal to g for every mass ( this is , of course, true in the vacuum where friction doesn't affect falling motion)

The correct expression for Newton's second law is ΣF = ma, or the sum of all forces is equal to mass times acceleration. On the left hand side, you have two forces, gravitational force and drag force. Drag force depends on the speed of an object, the air density, the cross-sectional area of the object and the drag coefficient. A feather will have a much greater drag force relative to its weight, so its acceleration will decrease more quickly than that of a hammer.  Rearranging the equation properly, you will have a = F_d/m - g, where F_d is drag force.

The force of gravity is NOT the same for the two objects. Remember Newton's Law of gravity:

F = GMm/r^2

The gravitational force depends on the mass of the object and the mass of the earth. 

Put this in for F in the equation a=F/m and the mass of the object (little m) cancels out. 

It is a very profound deep fact that the m in F=ma and the m in the gravity law are exactly the same thing. This is not a coincidence. It led Einstein to the general Relativity theory. 

The force is not constant with both objects 

f = ma 

F = Ma

IF the force of gravity increases with mass then (an only if) then F= mg

but F= ma  -> mg = ma -> g = a  The observed FACT that objects fall at the same rate is the reason we knew that the force of gravity MUST be proportional to its mass.

Yes, a = f / m.  With the feather and the hammer, the masses are different.  So far, so good.  But the forces are NOT the same.  The force of gravity is "per kilogram" (or other unit of mass). It is the ACCELERATION of gravity that is the same for all objects on the surface of the Earth.

Force[feather] = G * mass[feather] * mass[Earth] / R^2

R = distance to center of Earth.

Force[hammer] = G * mass[hammer] * mass[Earth] / R^2

You should be able to see that if Force is divided by the mass of the feather (or the hammer), we get a = G * mass[Earth] / R^2, which does NOT depend on the mass of the feather or hammer.

The force of gravity is proportional to mass.

This is more intuitive when you think about the larger object (e.g. the Earth) whose gravity is pulling on the smaller object. More mass -> stronger gravitational force.

But in fact gravity is proportional to the masses of both objects that are pulling on each other: 

F = GMm/d^2

where G is a constant and d is distance between the two objects.

To calculate acceleration, the mass m of the small object cancels out.

a = F/m = GM/d^2

Hence the constant g = (G)(mass of Earth) / (radius of Earth)^2 is the acceleration caused by gravity of any object near the Earth's surface.

In contrast, air resistance depends on an object's area, not mass. Air resistance exerts about the same force on both a hammer and a feather, which mean it decelerates the feather more than the hammer.

The force is mg, so substituting in your equation you get

a = mg /m, 

and the m's cancel to leave a = g, in other words the acceleration = gravitational acceleration, which is the same for any mass.

Where did you get the force being the same?  The acceleration is the same, and that's it.