It looks like I have just happened to find an easy reproducible solution for infamous:
'Load operation failed for query 'GetUser'.
The remote server returned an errror: NotFound'
issue for WCF RIA Services (Silverlight 5) web setup: when using VS2012 Web Publishing Wizard with 'Precompile during publishing' option checked-on then the issue does raise its ugly head. When the 'Precompile during publishing' option is checked off then deployed WCF RIA Services works well with Silverlight client. Please check on your system.
I have used fuslogvw etc.etc. - nothing helped. Never used Web Publishing Wizard before - xcopy was my friend, this time I wanted to automate the whole deploying process - and lost a couple of hours of precious time.
I still don't know what is the cause of the issue but I can proceed with my work. Next time when there will be more free time I will probably use this technique to localize the causes of the subject issue.
Edit for clarification: I have effectively solved the subject issue on my localhost but I have it still appearing in one program but not in another when releasing on an external web hosting environment, and I'm yet to find what causes this issue - do you know any good sources where this issue's various solutions are classified and accompanied with solution walk-throughs?