If you type "BLUE", in what order would those files be checked and/or run?

That is an interesting scenario, since you say you are in the BLUE directory, you would be working in a DOS environment.  If we look at computing rules, the BLUE.BMP wouldn't be used at all as it is a graphic file.  

 

When you type just a file name in DOS, the computer assumes you are trying to start some kind of program.  It would therefore look for any file ending in BAT COM or EXE, as these are all similar in their function.  Since computers are always sorted alphabetically ascending, (a to z), the BLUE.BAT file would be started. 

 

If you did have such a scenario (although highly unlikely) and you wanted the BLUE.EXE file to run, you would type the exact name including the extension and then hit Enter key.

 

The same rules apply if you use Run from the Start Menu, except you would have to type the complete path as well in the Open field (c:\BLUE\BLUE), because as stated previously, Run looks for executable files in the Windows directory.

 

I hope this helps clarify your question.