Any chance insurance would cover eyelid surgery ? Should I book medical or cosmetic consultation? (photo)

Coverage for eyelid surgery is dictated by your Insurance company and the terms of your specific policy. You will need to see an Oculoplastic surgeon who can examine you and perform the necessary testing and then submit this to your Insurance company for predetermination or pre authorization.

That's a maybe--usually a visual field test is required to determine if the skin hanging is compromising enough of your vision.  An ophthalmologist usually makes this determination.  But even if this is the case, corrective surgery would not likely focus on the asymmetry (the cosmetic aspect of your concern).  You might want to consider trying something simpler, like cosmetic botox to address the muscle that pulls down on the brow.  More units could be injected to the right side, where the brow is lower, in an effort to create symmetry.  It might solve the problem in a temporary way--but needs to be re-injected about every 4 months.  If you aren't happy with the results, could always consider surgical options like browlift or upper lid blepharoplasty.