What is the gap on the outer corner of my eyes and how can this be fixed? How much will it cost?

When the eyelid turns in, it is called entropion.  When the eyelid turns out, it is called ectropion.  This eyelid neither turns in or out.  It simply gaps away from the eye surface so it is accurately described as either a malposition or some might want to call it an ectropion.  Lateral canthoplasty is a surgical procedure used to correct this.

It is difficult from the pictures to tell exactly how loose your lower eyelid is, but if, after examination, by a specialist, you want it fixed, it can be closed and tightened by a procedure called a canthoplasty.  An ex[experienced Plastic Surgeon or Ophthalmologist should be able to do this for you, if indicated