The child was first referenced in X-Men: Messiah CompleX #1. She was not named Hope until Cable (2nd series) #10, after her late adoptive mother. Her birth name was revealed in Uncanny X-Men (1st series) #526.
Hope's childhood contains a number of missing years. According to Cable's narrative count, Cable (2nd series) #12 is shortly after Day 1543 of his efforts, making Hope roughly four years old. This fits with the stated two years they spent in New Liberty, #7-10, and the two years since her mother died mentioned at the start of #11. However, the narration in #11 also began referring to Hope as seven years old. Issues #11-12 led directly into Messiah War, which covered slightly more than 32.5 hours. Immediately after Messiah War, in #16, Hope is referred to as NINE years old.
Hope was specifically designed by the Phoenix Force to restart mutant evolution on Earth. Following AvX #12, mutants no longer require Hope's touch to manifest their abilities. It's unknown if the subliminal connection between Hope, her Lights, or the mutant population in general remains.