"Is hardness a index of materials against (plastic) deformation?"
Yes, and different indentation hardness tests impose different penetration geometries so give different results.
"Unlike the yield strength, the elastic modulus which are the fundmental properties of materials,..... The question is how to relate hardness with those fundmental properties? "
There is no particular reason they should be closely related. Elastic modulii are based on extremely small, recoverable deformations while indentation hardnesses are based on non-recoverable macroscopic deformations. These are completely opposite ends of a deformaton spectrum.
Consider the effect of heat treatment on a carbon steel. The elastic modulii don't change very much but hardness can be changed tremendously.
Probably the most that can be said regarding the relationship between modulus and hardness is something like a material's maximum possible hardness is proportional to its shear modulus - unlike the minimum hardness which is probably bounded by zero.
There is no logical relationship between a crystalline material's bulk modulus and hardness; When a crystalline material "yields" under hydrostatic loading it does so by changing to a different crystalline structure - it ceases to exist.
Dave