Whether the force is able to cause displacement or not depends on the mass m.
This is incorrect. A force will always attempt to accelerate an object, regardless of its mass, which causes displacement. The real factor here is whether friction is present. If the applied force is smaller than the static friction, the block will not move. If the applied force is greater than the static friction, the block will accelerate with an acceleration less than the applied force as friction in the form of kinetic friction will oppose the applied force.
Now if a second force F2 acts on the block vertically downwards such that it comes to a rest
Again, this is incorrect. The block cannot come to a rest if the two forces are the only forces acting on the block. When $F_2$ is applied, it will be opposed by an equal and opposite normal force by the surface. I think that you are again talking about friction. The formula for friction is $$F = \mu N$$ where $\mu$ is the coefficient of friction and $N$ is the normal force. So by pressing down on the block, you are increasing the friction, presumably until it has a magnitude greater than $F_1$, causing an acceleration in the opposite direction, until it comes to a stop.
person perceives the mass of the body to have increased
The mass of a body is an intrinsic quantity, and always remains constant. You are confusing mass with weight. Weight is the gravitational force acting on the block. Since the weight of the block is perpendicular to the acceleration, I do not see why either quantity will increase. The only quantity that changes is the total downward force acting on the block which is simply the sum of the weight and $F_2$. This is, of course, opposed by an equal and opposite normal force from the surface.