Contrary to what some people say, you CAN use salted butter. I use salted Kerrygold butter and it gives the final product a slightly salty flavor and a grainy texture.
Ghee is shelf stable so you don't have to put it in the fridge. You can if you want, but there is no point. It should last anywhere between 3 months and 1 years in an air-tight container, but try to use it as soon as possible for best flavor.
The quality of butter you use GREATLY effects the end result. Normal cheap store bought butter which looks almost white, has no real flavor and you'll notice it's probably a lot more water. Try to find a grassfed butter for this application.