What are my options to fix asymmetry of mouth? (photos)

Fillers and botox can be used to improve this area. I suggest seeing a facial sculpting expert.  Best, Dr. Emer.

A very common issue brought up by patients is concern over facial asymmetries.  It is important to understand that it is much more normal than not to have one brow higher than the other, and/or slight difference in height of the corner of the mouth.  Usually this is congenital, but sometimes is caused by conditions such as Bell's Palsy or trauma. That having been said, use of toxins such as Botox and fillers such as Juvederm, Restylane, or Radiesse can help minimize such asymmetries. In the photos shown, there is no evidence of paralysis, but instead a lower right oral commissure.  The minimally invasive treatments mentioned work very effectively for this in experienced hands.  

 Your smile appears crooked because of two reasons. First, your lips are different sizes and second, your muscles are pulling slightly different. You can manage both these problems with the combination of Botox to your DOA (depressor Oral angularis) a muscle. You can balance the difference in your lip size with a number of HA fillers. Obviously this should be done by an experienced doctor.