First of all, you need to understand what freefall motion is. A freefalling object has its motion perpendicular to the ground. The object has no initial velocity and encounters no air resistance. And yes, freefall motion is independent of the mass of the object.
Second, in your example, mass is not the reason why marker falls to the ground faster than paper. I can give you one thought experiment to disprove this idea. Imagine two people of equal mass jumping from a plane. Person A jumps with a parachute open, while person B has no parachute. Person A obviously has more mass the his counterpart, but he falls slower in comparison.
When an object falling with air resistance, the aerodynamics shape of the object dictates its vertical velocity. One criteria of this is the area cross section of the object. In your example, the paper has a much larger area which encounters air molecules compare to the marker, so it encounters larger air resistance or draft. If you have difficulty understand this, just imagine air molecules slowing the objects down by hitting them from the bottom. You can use this to understand why two object with different mass but identical shape fall to the ground at the same time from the same height.
And the problem with the penny is that it is not in freefall motion, again, because of the effect of air resistance. The penny is not going to keep on accelerating forever. Imagine you are walking in a swimming pool. Water in front flows around you to the back, but when you walk faster, you'd feel greater and greater resistance from the water because it has less time to flow to the back of you. Once you reach a certain speed, water cannot get out of the way fast enough. Thus, it compresses in front of you, and you can't go any faster than this speed.
The same thing happens to the penny falling through the air. Once it reaches a certain speed or "terminal velocity", it can't go any faster. So, it continues to fall in a constant speed until it hits the ground. The terminal velocity of the penny is around $40km/h$ after falling for about $15m$.
Now, when penny or any other objects that are "supposedly" drop on your head, the damage is not only depend on the velocity when it hits, but also the mass of the object. This is the concept of kinetic energy. Kinetic energy is given as $$K=\frac{1}{2}mv^{2}$$ When the penny hits you, its kinetic energy transfers to your head. Since penny is very light and not aerodynamic, it has little kinetic energy when it falls onto you. Other objects, depend on their mass and shape, can surely ruin your days if one drop on your head from high enough.
