First some background: There is a lowest energy equilibrium distance for each bond between molecules in the spring. When a string is stretched or compressed, the molecules are pulled or pushed (respectively) to distances of separation other than this equilibrium distance. This stores chemical potential energy in the bonds between the molecules of the spring. The restorative force of the spring is caused by the sum of all the restorative forces of the individual bonds between molecules in the spring. See the answer by "Farcher" in this thread for a more detailed treatment: Why is the restoring force directly proportional to extension?
As to your question. While the sum of all the restorative forces between molecules in the spring is directed opposite the spring's displacement from equilibrium, the restorative force of each individual bond that is stretched points in a direction determined by the molecular structure of the spring, not necessarily the exact opposite direction of the spring's overall displacement. Therefore some of the chemical potential energy stored in the bonds of the spring fails to be returned as kinetic energy and instead causes vibrations on the molecular scale not associated with the macroscopic oscillations of the spring. Temperature is a measure of the "random" vibrations on a molecular scale in a medium, therefore some of the chemical potential energy stored in the spring increases the spring's temperature when it is released. This corresponds to a lessening of the spring's amplitude of oscillation by conservation of energy.
In addition to this, as you mentioned, the spring may lose energy to drag as it passes through a medium (such as air), but even in a system with negligible drag, the above process still accounts for a loss of mechanical energy as the spring oscillates.
The rate of this loss of mechanical energy depends on the molecular structure of the spring. In fact, any solid has this property, but the things we call "springs" have molecular structures such that a significant portion of the chemical energy stored in its bonds as a result of stretching or compressing is returned as mechanical energy upon returning to equilibrium.
If the spring moves "more" unit per time (greater amplitude or frequency), then you will lose more mechanical energy. I am not sure how changes in the spring constant would effect the rate of loss of mechanical energy because the way to change the spring constant is to change the molecular structure of the spring.
This is basically the same question / answer and cites a paper which probably goes into more detail (that I haven't read):
Friction between atoms in spring
Hope that makes sense!
I have been doing some research about the affect of friction on the acceleration of such system. 
