That's very strange. What O/S and dosbox version? I'm sure someone can send you the exe if need be anyway.

I tried that! I use WinXP (SP1) and DosBOX 0.63 but I tried also the other versions of DosBOX! A friend sent me the DosBOX.exe as a ZIP File and when I unpack the File there is still no DosBOX.exe!

Do you see any dosbox file with no extension? If so, perhaps it really is dosbox.exe. Just a thought... The Windows default is set up so that extensions are hidden. It's difficult for me to believe that the other files would extract but the exe wouldn't. If that's really the case, then maybe you're out of diskspace. Another possibility for continued failure is maybe that you tried to extract the exe to a folder which you don't have permissions for. Try extracting to your desktop first. That way, you'll at least see a new file appear immediately (which you can later move to the appropriate folder).

No, I can't see any file without extension! There are really only 13 Files in the Folder! I have administration rights for my Computer, so I have permission to write to every folder! I tried to extract it to another folder, but it doesn't work!

Might be worth doing a virus/malware scan.
Also, are you running a virus scanner? If so, maybe it's misdetecting the dosbox.exe as a virus and deleting/quarantining (moving) it?