Last night I put a batch of wine going (very simple setup - a plastic bucket with a sealed lid and a cork in the top for a plastic airlock). This morning, I can tell the sugar has settled to the bottom (you can feel and hear it swishing around the bottom if you pick it up and move it a little). Should I open the sealed lid to stir it all up again, or leave it alone?
You can absolutely open the bucket if you feel it's necessary to stir the must. There is very little chance of contamination if you are diligent in sanitizing everything that will touch the must. If any air borne particles do get in there won't be enough to get a foot hold and will be overtaken by the yeast.
I have made over 20 meads that I have removed the lid every day for the first 7 days of fermentation to stir and to add nutrients and have never had an infection. I also do open fermentation where I don't even put a lid on the bucket.
I think there is a difference in being careful and being paranoid. I was once paranoid myself but with experience and knowledge from others I have moved on to careful.
I would let it sit for now. Sugar tends to dissolve eventually and yeast are crafty, so they will find the sugar at the bottom and eat it, regardless of whether it has dissolved, although your initial gravity readings may have been off if the sugar was not dissolved when you took them.
You want to be really careful right now to avoid infection since there is no alcohol yet to prevent it. If you are really concerned, the most I would do is shake or swirl the bucket to stir up the sugar and try to dissolve it that way, but I wouldn't recommend opening it.