1) In GSX Settings-> Audio-> Devices dropdown, My headphone don't show
Be sure they are working before you start GSX. If they connects after you start it, you must restart GSX after they are connected.
2) When I click on the GSX icon in the Toolbar, The GSX menu appears and disappears with 10 second. Is there a way to keep it open until I close it?
Each menu has different timeout times depending on the context, so it won't take extra space on screen for more than strictly necessary.
Some menus have timeouts of 30 seconds, so you have plenty of time to decide what to do, others like the operator selection have shorter timeouts with a default choice made automatically at the end of the timeout, so you can use it "hands free" if you are ok with the operator being selected automatically.
The menu can't be pinned on screen. If you think there's a particular menu that has timeout too short, please make an example.