First off, Psychopathy and Sociopathy are not the same thing. And you have them backwards. Sociopathy is based on actions and Psychopathy is based on more abstract thoughts, like the inability to empathize. Secondly, Empathy is simply the ability to understand the emotions and feelings of others. Not that you have to have gone though such a thing.
So no. Wrong on both counts.
Sociopathy is not specifically a thing anymore. It has been subsided into Antisocial Personality Disorder, which constitutes wholesale rejection of the social contract. Specifics vary from case to case, but generally involve impulsiveness, unjustified violence, and an inability to coexist with other humans.
Psychopathy is the most terrifying shit you will ever read about. Psychopaths, as you stated, do not feel empathy, but that isn't all. Psychopaths do not experience any emotions in the same way other humans do. They experience what are called shallow emotions, and the name really says it all. Shallow emotions are situational and fleeting, generally representing some short-term physical desire of the psychopath, such as food or sex. They do not last but are very compelling to psychopaths, who lack the kind of perception to reject them easily.
Psychopaths may outwardly appear normal, and most that have survived to adulthood "undetected" will almost certainly be. It is common for psychopaths to identify that they are fundamentally different from most other human beings in their early childhoods, and the psychopath in question will usually take steps to emulate normal humans on the surface of their interactions. Less proficient psychopaths will fail to do this and be detected almost immediately due to their uncontrollable and disturbing behaviors.
This acts as a sort of filter for psychopaths, where only the ones capable of social manipulation will remain under the radar for years. Thus the perception that all psychopaths are the kind of apparently suave but ultimately sadistic individuals that some of them are arises. These psychopaths are capable of creating multiple facade personalities to appease and entrap others so that they may indulge their desires with lessened threat of intervention from non-psychopaths. You could know a psychopath for a long period of time and never know the truth.
Psychopaths do not feel guilt, stress, or empathy and will lie pathologically in any situation. Psychopathy is considered to be part of the "Dark Triad" of personalities, those being personalities that are socially destructive. The other two are narcissistic and Machiavellian personalities.