I watched a couple segments of an anime a few months ago and lost the name somewhere. In the story Japan is under attack and has been for some time. The attackers appear with some kind of time /dimension quake that leaves a large crater. In the crater appears a female with some kind of power. A schoolboy approaches one and prevents more damage by trying to befriend her. A government military group starts a program to have him meet all arrivals and befriend them to stop damage. One early character is a small girl with a rabbit costume that has large droopy ears with buttons on the end. She carries a stuffed rabbit that has an eye patch over one eye. I would really like to get the name of this one.
According to your description the anime you are searching looks like Date A Live.
Synopsis from MAL
Itsuka Shido is a high school boy. On the last day of the spring holidays, an explosion destroys the town and a girl in armor appears in front of him. Shido's sister-in-law, Kotori, tells him that the girl is called "Special Disasterous Designated Creature: Spirit", which causes a timespace quake. Kotori disclosed that she is the commander of the anti-spirit organization Ratatosk and orders him to go out on a date with the spirit girl. Kotori says, "I don't ask you to beat the spirit. Just let her fall in love with you and save the world."
And the small girls with bunny costume is Yoshino.