It only has Shader Node icons for now, so it doesn't work with Geometry Nodes or any other node editor types. I hadn't planned on making icons for those editors, but enough people have expressed interest in Node Bar for Geo Nodes, so I will be working on that within the coming months.

No, Node Bar does not run on blender startup and it has to close when the Shader Editor is closed. Other than that, it will remain open until you close it.

I recommend setting a simple hotkey for Full Bar and MiniBar in the Add-on preferences. That way it can be toggled on and off quickly. The defaults are A for Full Bar and B for Mini Bar.