I've just purchased ghee/clarified butter for the first time. It was on the shelf when I bought it. Do I need to refrigerate it once I've ope

It depends on the ghee. When I make mine, I let it go until the milk solids are very dark brown so my ghee is very flavorful. I would use it for seafood in a heartbeat. I dipped my steamers in it just recently. Taste your ghee and see if you like the flavor. If you do, you will like it melted.

It last longer in the 'fridge, but it's easier to use at room temp. Use it often instead of cooking oil for sauteed veggies as it has very high smoke point. It really helps shrimp and scallops have a great sear on them when you use a cast iron pan.

Learn to make it---it's super easy and you can control the 'nuttyness' of the flavor. All you need is cheese cloth to strain it. (or a very fine strainer). The 'net is full of instructions.

I make my own ghee and always keep it on my counter. No need to refrigerate it. When it gets super hot here I do only because it goes back to liquid form and it's easier to deal with in semi-solid state. The only problem with using it for a wine sauce is that the dairy is removed, so it won't emulsify into a cohesive sauce. I usually grab whole butter to make a sauce.

No, you can keep at room temp for a few weeks then pop into the fridge where it will last about a year.

And I can't see why you couldn't use it in a white wine sauce.