Hi, maybe I can clear up some questions here.
No, I don't think that's a mole, I think it's a spot of kajol, or kohl, that Indians often put on the cheeks of babies (or young women) to serve as a "flaw" so the otherwise unblemised beauty of the child (or woman) will not attract jealousy/the evil eye. It's a very common thing.
Sounds like they're south Indian. In some South Indian communities, the wife and child do not take the husband's surname (which is his father's first name); instead, they take his first name as their surname. Oftentimes as well you will find that the person only uses his first name in full, with the initial of his surname preceding it (e.g., John Smith would be S. John).
Regarding some earlier comments in this thread -- as south Indians, these people will not be from anywhere near Delhi. But as someone who has spent a lot of time in Delhi, I want to lodge my objection to any implication that kidnapping of children is routine or usual there! Were a case like this to happen in Delhi, it would be lighting up the newspapers there and shocking people just as much as it shocks us here.
My best wishes for Saanvi and her parents!