Why direction of stress is not pointing in opposite to that of external applied force? Stress describes how a body responds to an external force. Or, i.e., stress quantifies the internal force that resists the applied force to maintain a state of equilibrium. So stress, as a vector quantity, should point in the opposite direction to any externally applied force.
In the drawing below, if my hand applies an external uniform force pointing into the wall (or pushes against area A of the rod), the rod would compress and store potential energy to bounce back to its uncompressed state if my hand is off the rod. Therefore, the direction of stress intrinsic to the rod itself should point in the opposite direction, or out of the rod, under compression.
In textbooks or tutorials I found they always portray the directional arrows of "negative compressive stress" to be pointing into the rod), and "positive tensile stress" pointing out of the rod, which make absolutely no sense to me. To cause compression, external force must points into the rod which results in a stress pointing out of the rod. Vice versa for tensile force, if it pulls or stretches the rod, the force will induce a stress pointing into (the center of) the rod.
[Edit]
Let's ignore gravitational force here as that will bend the rod causing both tensile and compression stress and unnecessarily complicate the concept here.


 A: Your diagrams are showing the directions of the external applied compressive and tensile stresses. In order to show the directions of the resulting internal compressive and tensile stresses you need to draw free body diagrams of cut sections of the beams. Those diagrams will slow the direction of the internal stresses as opposite to the direction external stresses as you say they should be.
Hope this helps
A: Stress is actually a 2nd order tensor, which, by its very nature is bi-directional.  The usual sign convention is that compressive stress is negative and tensile stress is positive.  If you want a more precise description of how this all works, Google "Cauchy stress relationship," which provides the mathematical framework for mapping the stress tensor into the traction vector on a surface of arbitrary orientation within a material (or at its surface).
A: There is actually a very simple reason that compressive stress components are negative and tensile stress components are positive. For an elastic material the stress is proportional to the strain. This is Hooke’s law. A negative strain is the object getting smaller which happens in compression. So compressive stress components must also be negative so that the stress can be proportional to the strain
Now, regarding the direction of the force. It is important to understand that stress is not a vector, it is a tensor. A tensor is an object that if you give it a vector it returns another vector. In the case of stress, you give it a vector representing a small area and it returns the force on that area. The direction of the area vector is normal to the area and the length of the vector is the size of the area.
Now, at a given point in the object you can look at forces on areas pointing in any direction, but it is sufficient to consider the forces on the six faces of an infinitesimal cube. The thing is that each face of the cube has two sides, the inside and the outside. In principle, either could be chosen as the direction of the area vector.
Making tensile stress components positive corresponds to choosing the outside face as the direction of the area vector. With that, a positive stress component corresponds to an outward force which causes a positive strain. This is the force acting on the infinitesimal cube, not the 3rd law reaction force that the cube exerts on the rest of the material. It is a force internal to the object, but it is the force on the infinitesimal cube, not the reaction force as you described.
A: Thats simply the way compressive and tensile stress is defined. Stress isn't strictly a vector, having the same dimentions as pressure. So 'stress produced when the body is compressed' is called compressive stress, and 'stress produced when the body is stretched' is called tensile stress, even though as you said, the actual reaction forces are opposite.
