So pokemon that hatch from Eggs come at lvl 1. Their move power? Is it just the base power without counting stats? Wrong. Even a newborn's stats count but in a different way. Depending on the stat rating of the pokemon, the moves it has upon hatching will have varrying power. If a stat has a + then attacks benefiting from that stat will have +2 to the base power upon hatching. If a stat has a ~ then the base power is simply the base power of the attack uneffected by the stat. If a stat has a - then attacks from this stat will have -2 from their base power. Easy and short.
As far as hatching eggs, eggs hatch depending on how many real time days have passed since you recieved it.
Getting eggs, too, has changed. Rather than getting them from the daycare, you will recieve eggs from the Pokemon Center from the new feature, Fertila, an assistant to Nurse Joy, whose partner is Happiny. Speak with her every now and again in the pokemon center. Occasionally she'll have an egg without a home. However, occasionally, she will ask you to deliver the egg to someone for her. Delivering eggs is always rewarded with Some Poke and a Berry of some sort.