I have just read about 10 pages worth of explanations of time paradox and what Callie was doing with the photographs in the Time Lapse. However, not a single one could clearly answer the most basic question - could Callie actually change past and save Finn using the machine? If so, how? Considering machine always sent FUTURE photographs?
From what we see in Time Lapse, I'm pretty sure that time is immutable. The characters kept trying to change the outcome once it started going south, but because their information was limited to a single photograph they had to guess at the events leading up to the picture, which was often wrong and ultimately caused the picture to be correct.
Callie wanted to use the machine to change the past by sending a message to her past self. The problem is, in an immutable timeline, this is impossible. The future has already been written, which is why it can be photographed in the first place. She had already seen the photograph and it had no message. Therefore, she never sent it.
Note that this is similar to the way time works in 12 Monkeys. Cole, try as he might, cannot change the events leading up to the catastrophe that had befallen the Earth. He can't change the past, because the timeline is unchangeable. The best he can do (and his actual mission) is to collect information and possibly a pure sample of the virus so that the people of his time can find a cure.