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.

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.