Can you just get one lazy eyelid lift?

Of course. Of course but patients with one lazy eyelid will commonly have a lazy eyelid on the other eyelid that is not evident due to compensation for the more severe eyelid (herring reflex). Commonly treating the more diseased eye will result in un-masking the less severe eye. The eye surgeon will typically tape the more severe eye during exam to expose this problem.
Yes. But for perfect symmetry many times both eyelids have to be worked on.
Droopy eyelid. There are several causes of a droopy eyelid. Your best bet would be to have your eyes evaluated by a board certified facial plastic surgeon.
Yes. Yes, one droopy or lazy eyelid can be lifted. It is common, however, for the operating surgeon to do both eyelids at the same time as it may be easier to "match" the height of the eyelid after surgery that way. Discuss your options with the surgeon. Best choice is an "oculoplastic" surgeon, in my opinion.
Yes. You can just fix one side. Often, though, that will unmask issues on the other side - so your doctor must check for that possibility.