It is true in they right patient that a sliding genioplasty can create the functional improvements that you have described. However, such outcomes are more predictable in the more severely chin/jaw deficient patient that what you are. But it may be effective for you as well. It would require more pictures from different angles for a more complete assessment.
The photo demonstrates a minor recessive and weak chin profile, for which a small Silastic chin implant can improve and help with facial balance and proportions. A chin implant will not help with the way the teeth fit together. Breathing difficulties out of the nose can be caused on a variety of issues such as a deviated nasal septum, turbinate hypertrophy, valve collapse, and allergies. Each one of these issues are treated differently with either medications or surgery. For more information and many examples of chin implants, please see link and the video below
On profile, your chin is minimally recessive, at best. Doing a genioplasty is abdolutely not indicated in this case. A relatively small chin implant will correct whatever disparity your chin profile presents.This will not change your breathing issues which may be due to something else. An implant will support a "hanging" lower lip.