Can I reset Home button sensitivity? I don't have a problem with the button itself, it seems that I need to push deeper to make it work.  I don't have to push harder... just deeper.

I remember someone talking about the "fact" you can reset sensitivity for the Home button of the iPhone.

Does this software sensitivity reset really exists ?
No, wrong. The home button is no sensor in the meaning of a pressure sensor.

For reference see this.