Wow, only just saw this!
Okay, as far as I understand, a portrait has to be charmed in some way in order for it to be able to talk. I suspect that, as ALL the portraits at Hogwarts seemed able to talk, there was some kind of castle-wide charm in effect.
I have always assumed that photographs are something different and can only move, not speak - and they can be taken on a muggle camera and developed by a wizard and still move (as we saw with Colin in the second book).
Dumbledore's portrait, and the other headteachers: I kind of imagine them as a special case, a little bit like a benign horcrux in that they contain some part of the personality of the person they represent. That's not to say that portraits like the Fat Lady don't have a personality as well, but perhaps once a witch or wizard has been head for some time, a small part of their knowledge remains after they've gone. Personally, I like the idea that as the old head dies, they kind of appear in a blank portrait frame (have you seen Stardust? I'm thinking of how the brothers die and then reappear as ghosts a few seconds later, that kind of thing). I'm not sure if I'm entirely getting across what I'm trying to say here!
As for Carlotta painting someone - certainly she would be able to charm the portrait to move (Sam does this with his paintings all the time) but I doubt she would be able to have the picture talk or react any more than smiling and waving (and now I'm thinking of Madagascar... xD)
Once again, sorry for the delay in answering this, hope it helps a bit
Carys x