A hardened steel goes through a heat treatment process that reaches very high temperatures until it is quenched in oil, usually. Then it goes through another tempering process. This results in really high hardness, stiffness and overall strength. An annealed steel goes through a heating process but it's then cooled slowly. This causes a bit of relief in the crystalline structure. It becomes more ductility and less hard than untreated steel. The E of a hardened steel could rise from 200GPa to very high 700GPa while that of annealed steel stays around the untreated range depending on the alloy. 200-300GPa.

Mostly it remains the same. Consider Stress-Strain diagram. Youngs Mod (E) is only in the linear region. Beyond the linear region, we can not apply the value of E to solve for nonlinear behaviors. Now consider Heat treatment processes (They can raise or lower the yield strength, UTS, etc of a material after heat treatment process) but the Linear region will still have the same slope as untreated material.

Same