Who took the pictures: For "Cold War" it was likely just some crew member on the submarine. For "HIde" it would most likely be the time traveler. Though, I think it was the Crooked Man.
As for why the kids saw the pictures? They said they saw them in school, so they were likely learning about the cold war, and they were just looking at photos of Russian submarines crews.
After seeing their nanny and her "boyfriend" in a picture that was taken in 1983, they got curious and tried to learn what was up. That guy from "Rose" found dozens of pictures of the Ninth Doctor throughout the decades, so them finding a few pictures of Clara in different time periods isn't too difficult to grasp.