That, of course, changes everything. Sorry, Emily, but your parents are criminals. You can be re-united with them when they go back to Guatemala where they belong.
As for why Emily was sent back, it's because she would otherwise be an unaccompanied minor, and the airlines won't deal with children that young as unaccompanied minors. About the youngest that they will deal with is typically about 8 years old. ICE probably should have given her parents an opportunity to come and pick her up, but then there also would be the problem of proving custody. Lionel probably doesn't have any good ID on him, so how would he prove that he was her father? They can't just let anyone take the child who is in their custody at the moment, so my guess is the only thing that they could do with her was send her back from whence she came.