How do I modify the mesh gradient of a stroke and then switch and modify the mesh gradient of the fill as well? I've been having to duplicate the object and modify the mesh there and then change the fill/stroke on the original once I'm done.

You have to switch between Fill and Stroke properties, but mesh gradient is applied simultaneously and can be adjusted by the Node tool:

When I do that, though, the nodes don't show up for the mesh gradient. I have to use the mesh gradient tool first or else it won't work.