You have a large soft chin pad which becomes protrusive when you smile. That can be improved by a submental resection of some of the chin pad and redrawing over the bone.
The chance of an asymmetry following genioplasty is minimal. Most surgeons use a method to mark the midline of the chin and use this measurement to keep the jaw aligned. The pre-stepped plates used today also help maintain the chin segment centered. Lastly, the soft tissue or skin and muscle ...
Thank you for your pictures. Give your results 3 months to mature. Once all of the swelling goes down you should noticed a more refined chin contour. Best wishes