When I completely relax my face my eyebrow and eyelid droops. Would surgery help? (Photo)

As an over the internet answer I might try BOTOX to elevate the brow and a skin only upper lid blepharoplasty. 

A brow lift may be more involved than you need at your young age.  You might consider Botox or Dysport to the space between your brows as well as under the tail of the brow to see if that relieves some of the hooding.  If this isn't dramatic enough, then you might consider an upper eyelid blepharoplasty.

You have some brow asymmetry which contributes to the different look from one side to the other.  Have you have ever had botox treatments before?  Do you sleep on your right side?  Where you born this way?  I would recommend a consultation with a facial or oculoplastic surgeon to determine which is the best way to proceed.