What is wrong with my eyes and how can I help it? (photo)

When people say someone looks Chinese, the mean that they have a full upper eyelid fold with a small amount of upper eyelid platform showing.  You of course have a second unrelated issue that has nothing to do with looking "chinese" (you don't).  Your lower eyelid is not properly positioned against the eye.  There is a gap that should not be there.  This can and should be corrected to ensure that tears are propperly move on the corneal surface.  See an oculoplastic surgeon for this problem.  A lateral canthoplasty surgery is used to correct this.

It looks like you have lower lid malposition, and from your description this I likely congenital (aka you were born with it). This isn't what necessarily lends to an "Asian" eye though. You can undergo a procedure where the lower lid is repositioned, but bear in mind this will reduce the show of the white part of the eye on the sides. So your eyes may appear smaller.