Place Visual Studio resource files in a folder other than 'Resources'?

This is a little tricky, but it is possible.

VS checks if the file added to a resource is already defined somewhere within your project. If it can't find it, it creates the folder Resources, puts a copy to the file there, adds this file to the project and puts a reference into the resource designer to this fresh copy of your file.

To avoid this behaviour you should add the file to your project before you add it into the resource file. If the file isn't somewhere within your project structure you can just create a folder, right click it, select Add file and before you click on the Add button of the OpenFileDialog, push on the little arrow next to the button and select Add as link.

Now the file resides on the place on your hdd wherever you like and the resource designer doesn't create a copy within your project file if you now add the file within the resource designer.

Maybe this little picture helps to find the Add as link button:
_{(source: modbusdriver.com)}

That's just a subdirectory of your project directory. Your program doesn't use it at runtime, it should use the embedded resources. Anything you add to the .resx file gets copied there, not just images. But you can rename the folder if you really want to, right-click it and click Rename.

Instead of adding a .resx file to your project, I'd recommend you use the existing one. Project + Properties, Resources tab. Makes it very easy to retrieve the resource at runtime, just use Properties.Resources.Something in your code.