I guess I developed some empathy sometime...

But I don't think that was empathy we saw in the video. I think it was a bit of "monkey see, monkey do"--I'll bet that mommy was making a sad face while singing. Not to trivialize that; some child psychologists believe that the very first human learning is conducted through the simple transaction of mommy smiling and baby reacting. The important lesson comes when baby learns he/she can make mommy smile by smiling, and thus becomes a participant in the world instead of a spectator.

The mental faculty that makes empathy possible is called "the theory of mind", and starts developing around 7-9 months, but doesn't become fully-developed until 3 or 4. The wikipedia article first paragraph sums it up pretty well:

Theory of mind (often abbreviated "ToM") is the ability to attribute mental states—beliefs, intents, desires, pretending, knowledge, etc.—to oneself and others and to understand that others have beliefs, desires, and intentions that are different from one's own.[1] Deficits occur in people with autism spectrum disorders, schizophrenia, attention deficit hyperactivity disorder,[2] as well as neurotoxicity due to alcohol abuse.

The article's section on "Development" is pretty germane. This last paragraph connects that smile-transaction to theory of mind:

Recent research in developmental psychology suggests that the infant's ability to imitate others lies at the origins of both a theory of mind and other social-cognitive achievements like perspective-taking and empathy.[37] According to Meltzoff, the infant's innate understanding that others are "like me" allows it to recognize the equivalence between the physical and mental states apparent in others and those felt by the self. For example, the infant uses his own experiences orienting his head/eyes toward an object of interest to understand the movements of others who turn toward an object, that is, that they will generally attend to objects of interest or significance.

The "theory of mind" is, to me, a mental faculty, or power, or ability, or whatever; and like all human gifts, there's a wide variety in how strong and accurate it is in different people. I think sociopaths have a very weak ToM, and tend to think of people as just objects in the environment to be manipulated. And of course people we perceive as nurturing and caring have lots of Tom, and it's been observed that people like royalty have a gift for putting other people at ease. That is, "graciousness" might require empathy.

One of the dangerous things about sociopaths is that they tend to intellectually work out something like the theory of mind, a set of rules based on observation that lets them predict how other people will react and behave, and from that sociopaths can synthesize charm and psuedo-warmth. Sociopaths are manipulators based on that knowledge, but they're manipulating complicated machines, not people.

Gotta go finish the lentils...