Can gravlax be hung to dry?
I'd strongly advice against that. A cure only works as long as the salt and sugar concentrations are high enough. And although Gravlax does a decent job of infusing the salmon with both the content in the salmon itself is not nearly high enough to keep it preserved, it is the crust on the fish itself that keeps it ok to eat.

It is not a problem to keep your salmon under its crust for a week, just make sure to keep it on a rack so liquid can drain away.

What you could try is making a version of Stockfish but you'll need a dry and cool environment (like Norwegian cool). Stockfish is usually made with very lean fish, I don't know how that would work with fatty salmon. But a casual google seems to confirm people are making something called "salmon jerky" so it might just work.