I was just wondering if it's possible to move the dll and xml files into a subfolder, like in this picture:
This isn't really so much of a MonoGame issue as much as it is a .NET issue.
The XML files are unimportant and can safely be deleted as they only contain the documentation for the matching DLL files. I'm not sure if there is any kind of legal requirement for you to keep them around but I seriously doubt it. I am not a lawyer.
Moving the DLL files though will prevent GameName1.exe from locating them at runtime. This could possibly be fixed though I'm not sure it's really worth the effort. The rules for locating DLLs in the .NET runtime can be found here: https://msdn.microsoft.com/en-us/library/yx7xezcf.aspx?f=255&MSPPError=-2147217396. I think you can tell .NET to look for assemblies in another folder through a configuration file.