Eyelid surgery. Yes, there is a way to treat lazy eyelid on most occations. Some require medications (such in cases of miastenia) others are surgical. You need to see a plastic surgeon or occuloplastic sprecialist.
Yes. It's called levator advancement.
Yes. There are two types of "lazy eyelid"; congenital and acquired. Although they are corrected with different methods, both are correctable. You should consult an oculo-plastic surgeon for proper evaluation and to discuss surgical options.
Yes. A droopy eyelid is referred to as "ptosis" and there are various surgical methods of correction depending on the underlying cause and the severity of the ptosis. I suggest consulting with a local plastic surgeon for an in-depth discussion regarding the surgical options, risks, and benefits.